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Learn Sine and Cosine Rules in Trigonometry: Worksheets & Examples

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K

Kate Robinson

05/07/2022

Maths

Trigonometry for Scalene Triangles and Trigonometric Equations

Learn Sine and Cosine Rules in Trigonometry: Worksheets & Examples

Trigonometry rules help solve triangles by finding unknown sides and angles through specific formulas and relationships.

The sine rule is essential when working with triangles where you know either two angles and one side (AAS) or two sides and a non-included angle (SSA). The formula states that the ratio of any side to the sine of its opposite angle is constant for all sides and angles in a triangle. When learning how to use the sine rule in trigonometry step by step, students start by identifying known values, substituting them into the formula, and solving for the unknown. This rule is particularly useful in non-right triangles where the Pythagorean theorem cannot be applied.

The cosine rule comes into play when dealing with triangles where you know either three sides (SSS) or two sides and their included angle (SAS). The cosine rule formula for side is c² = a² + b² - 2ab cos(C), where C is the angle between sides a and b. When to use cosine rule depends on the given information - it's particularly valuable when working with obtuse triangles or when the sine rule cannot be applied. Students often practice with cosine rule questions and answers PDF resources to master these concepts. Bearings in mathematics add another layer of complexity, combining angle measurements with directional navigation. Bearings trigonometry examples with solutions typically involve real-world applications like navigation, surveying, and engineering problems. When working with bearings Maths questions and answers, students must understand both the three-figure bearing system and how to apply trigonometric ratios to solve practical problems. The combination of bearings and trigonometry creates a powerful tool for solving real-world spatial problems, from maritime navigation to land surveying.

These concepts build upon each other, with sine and cosine rule questions and answers PDF resources providing comprehensive practice opportunities. Understanding when to use sine rule and cosine rule is crucial for success in advanced mathematics courses and practical applications. Students often benefit from working through bearings with trigonometry worksheets that provide structured practice with increasing complexity, helping them develop confidence in applying these fundamental concepts.

...

05/07/2022

240

Trigonometry
The Sine Rule
A
PROOF
b
b
Examples
A
C
с
d
1720
P
a
76
1 find the length of the side x
C
750
10
C
the sine rule
(find sides!
an

View

Understanding the Sine Rule in Trigonometry

The sine rule is a fundamental concept in trigonometry that helps solve non-right-angled triangles. When working with triangles where you know two angles and one side, or two sides and one angle, the sine rule becomes an invaluable tool.

Definition: The sine rule states that in any triangle, the ratio of the length of a side to the sine of the opposite angle is constant. It can be written as: a/sin A = b/sin B = c/sin C

Understanding when to use the sine rule is crucial for solving trigonometric problems effectively. The rule is particularly useful when you have either:

  • Two angles and one side
  • Two sides and an angle opposite to one of them

When applying the sine rule to find angles, ensure you're working with the correct corresponding sides and angles. Remember that angles must be opposite to their respective sides in the formula. For accurate calculations, maintain precision by keeping intermediate steps to more decimal places than your final answer.

Example: To find a missing side x in a triangle where angle A = 75°, angle B = 72°, and side b = 8: x/sin 75° = 8/sin 72° x = (8 × sin 75°)/sin 72° = 7.89 units

Trigonometry
The Sine Rule
A
PROOF
b
b
Examples
A
C
с
d
1720
P
a
76
1 find the length of the side x
C
750
10
C
the sine rule
(find sides!
an

View

Mastering the Cosine Rule in Trigonometry

The cosine rule serves as an essential tool when working with triangles where the sine rule isn't applicable. This rule is particularly valuable when you have three sides of a triangle or two sides and the included angle.

Formula: For any triangle with sides a, b, c and angle A: a² = b² + c² - 2bc × cos A

The cosine rule is effectively an extension of the Pythagorean theorem for non-right-angled triangles. When to use cosine rule becomes clear in situations where you have:

  • All three sides and need to find an angle
  • Two sides and the included angle, needing to find the third side

Understanding the relationship between the cosine rule and Pythagorean theorem helps visualize its applications. When the angle is 90°, cos 90° = 0, reducing the formula to the familiar a² = b² + c².

Highlight: Always ensure your calculator is in degree mode when using the cosine rule, and maintain accuracy by keeping intermediate calculations to more decimal places than your final answer.

Trigonometry
The Sine Rule
A
PROOF
b
b
Examples
A
C
с
d
1720
P
a
76
1 find the length of the side x
C
750
10
C
the sine rule
(find sides!
an

View

Navigating Bearings in Trigonometry

Bearings represent directions measured clockwise from north, providing a precise way to describe locations and movements. Understanding bearings is crucial for navigation, surveying, and practical applications of trigonometry.

Vocabulary: Three essential rules for bearings:

  1. Always measure from north
  2. Measure clockwise
  3. Express with three digits (add leading zeros if necessary)

Working with bearings often involves combining trigonometric calculations with directional problems. When solving bearing problems, create clear diagrams showing:

  • The north line at each relevant point
  • The bearing angle measured clockwise from north
  • The distances between points

Example: For a ship problem with two bearings and distances, first draw the triangle formed by the positions, then apply the sine or cosine rule as appropriate to find unknown distances or angles.

Trigonometry
The Sine Rule
A
PROOF
b
b
Examples
A
C
с
d
1720
P
a
76
1 find the length of the side x
C
750
10
C
the sine rule
(find sides!
an

View

Advanced Triangle Area Calculations

Finding the area of non-right-angled triangles requires specialized formulas beyond the basic ½ × base × height method. These techniques are essential when perpendicular heights aren't readily available.

Formula: Area = ½ × a × b × sin C Where a and b are two sides of the triangle, and C is the included angle

This formula provides a direct method for calculating areas without needing to find the perpendicular height. It's particularly useful when working with:

  • Triangles where two sides and the included angle are known
  • Problems involving sectors and segments of circles

The relationship between this area formula and the traditional method becomes clear when you consider that sin C × b represents the perpendicular height when using side a as the base.

Example: For a triangle with sides 8cm and 10cm, and included angle 43°: Area = ½ × 8 × 10 × sin 43° = 27.3 cm²

Trigonometry
The Sine Rule
A
PROOF
b
b
Examples
A
C
с
d
1720
P
a
76
1 find the length of the side x
C
750
10
C
the sine rule
(find sides!
an

View

Trigonometric Equations

Solving trigonometric equations is an essential skill in GCSE further maths. These equations involve finding values of angles that satisfy certain conditions.

Example: Solve sin θ = 0.3 for 0° ≤ θ ≤ 360°

Solution:

  1. θ = sin⁻¹(0.3) ≈ 17.5°
  2. Due to sine rule ambiguity, there's a second solution: 180° - 17.5° = 162.5°

Highlight: When solving trigonometric equations, it's often helpful to sketch the relevant graph to visualize the solutions.

The page also covers equations involving cosine and tangent functions:

  1. cos θ = -1/2 Solutions: θ = 120° or 240°

  2. tan θ = 4 Solution: θ ≈ 76.0°

Vocabulary: Theta (θ) is commonly used to represent unknown angles in trigonometric equations.

Trigonometry
The Sine Rule
A
PROOF
b
b
Examples
A
C
с
d
1720
P
a
76
1 find the length of the side x
C
750
10
C
the sine rule
(find sides!
an

View

Advanced Trigonometric Problems

This section deals with more complex trigonometric equations and introduces trigonometric identities, which are crucial for GCSE further maths and beyond.

Example: Solve 6 sin θ + 5 = 0 for 0° ≤ θ ≤ 360°

Solution:

  1. 6 sin θ = -5
  2. sin θ = -5/6
  3. θ = sin⁻¹(-5/6) ≈ -56.4°
  4. Adjusting for the given range: 303.6°
  5. Second solution: 180° - 56.4° = 123.6°

Highlight: When solving more complex equations, breaking them down into simpler steps and using known trigonometric identities can be very helpful.

The page also touches on trigonometric identities, which are equations that are true for all values of the variables involved. These identities are fundamental in simplifying and solving advanced trigonometric problems.

Vocabulary: Identities - equations that are true for all values of the variables involved, regardless of what those values may be.

This comprehensive guide covers essential topics in trigonometry for scalene triangles, providing students with the tools they need to tackle a wide range of problems in GCSE maths and beyond.

Trigonometry
The Sine Rule
A
PROOF
b
b
Examples
A
C
с
d
1720
P
a
76
1 find the length of the side x
C
750
10
C
the sine rule
(find sides!
an

View

The Sine Rule

The sine rule is a fundamental concept in trigonometry for scalene triangles. It provides a method for finding unknown sides or angles in non-right-angled triangles.

Definition: The sine rule states that the ratio of the sine of an angle to the length of the opposite side is constant for all angles in a triangle.

The formula is expressed as:

a / sin A = b / sin B = c / sin C

Where a, b, and c are the lengths of the sides, and A, B, and C are the angles opposite to those sides respectively.

Example: To find the length of side x in a triangle with angle A = 75°, side b = 10, and angle B = 72°, we use the sine rule: x / sin 75° = 10 / sin 72° x = (10 * sin 75°) / sin 72° ≈ 10.3

Highlight: When using the sine rule to find an angle, there may be two possible solutions due to the ambiguity of the inverse sine function. Always check if both solutions are valid in the context of the problem.

Vocabulary: SOHCAHTOA is a mnemonic device used to remember trigonometric ratios in right-angled triangles, but the sine rule extends these concepts to non-right-angled triangles.

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Maths

240

5 Jul 2022

8 pages

Learn Sine and Cosine Rules in Trigonometry: Worksheets & Examples

K

Kate Robinson

@katerobinson_rafs

Trigonometry rules help solve triangles by finding unknown sides and angles through specific formulas and relationships.

The sine ruleis essential when working with triangles where you know either two angles and one side (AAS) or two sides and a... Show more

Trigonometry
The Sine Rule
A
PROOF
b
b
Examples
A
C
с
d
1720
P
a
76
1 find the length of the side x
C
750
10
C
the sine rule
(find sides!
an

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Understanding the Sine Rule in Trigonometry

The sine rule is a fundamental concept in trigonometry that helps solve non-right-angled triangles. When working with triangles where you know two angles and one side, or two sides and one angle, the sine rule becomes an invaluable tool.

Definition: The sine rule states that in any triangle, the ratio of the length of a side to the sine of the opposite angle is constant. It can be written as: a/sin A = b/sin B = c/sin C

Understanding when to use the sine rule is crucial for solving trigonometric problems effectively. The rule is particularly useful when you have either:

  • Two angles and one side
  • Two sides and an angle opposite to one of them

When applying the sine rule to find angles, ensure you're working with the correct corresponding sides and angles. Remember that angles must be opposite to their respective sides in the formula. For accurate calculations, maintain precision by keeping intermediate steps to more decimal places than your final answer.

Example: To find a missing side x in a triangle where angle A = 75°, angle B = 72°, and side b = 8: x/sin 75° = 8/sin 72° x = (8 × sin 75°)/sin 72° = 7.89 units

Trigonometry
The Sine Rule
A
PROOF
b
b
Examples
A
C
с
d
1720
P
a
76
1 find the length of the side x
C
750
10
C
the sine rule
(find sides!
an

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Mastering the Cosine Rule in Trigonometry

The cosine rule serves as an essential tool when working with triangles where the sine rule isn't applicable. This rule is particularly valuable when you have three sides of a triangle or two sides and the included angle.

Formula: For any triangle with sides a, b, c and angle A: a² = b² + c² - 2bc × cos A

The cosine rule is effectively an extension of the Pythagorean theorem for non-right-angled triangles. When to use cosine rule becomes clear in situations where you have:

  • All three sides and need to find an angle
  • Two sides and the included angle, needing to find the third side

Understanding the relationship between the cosine rule and Pythagorean theorem helps visualize its applications. When the angle is 90°, cos 90° = 0, reducing the formula to the familiar a² = b² + c².

Highlight: Always ensure your calculator is in degree mode when using the cosine rule, and maintain accuracy by keeping intermediate calculations to more decimal places than your final answer.

Trigonometry
The Sine Rule
A
PROOF
b
b
Examples
A
C
с
d
1720
P
a
76
1 find the length of the side x
C
750
10
C
the sine rule
(find sides!
an

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Navigating Bearings in Trigonometry

Bearings represent directions measured clockwise from north, providing a precise way to describe locations and movements. Understanding bearings is crucial for navigation, surveying, and practical applications of trigonometry.

Vocabulary: Three essential rules for bearings:

  1. Always measure from north
  2. Measure clockwise
  3. Express with three digits (add leading zeros if necessary)

Working with bearings often involves combining trigonometric calculations with directional problems. When solving bearing problems, create clear diagrams showing:

  • The north line at each relevant point
  • The bearing angle measured clockwise from north
  • The distances between points

Example: For a ship problem with two bearings and distances, first draw the triangle formed by the positions, then apply the sine or cosine rule as appropriate to find unknown distances or angles.

Trigonometry
The Sine Rule
A
PROOF
b
b
Examples
A
C
с
d
1720
P
a
76
1 find the length of the side x
C
750
10
C
the sine rule
(find sides!
an

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Advanced Triangle Area Calculations

Finding the area of non-right-angled triangles requires specialized formulas beyond the basic ½ × base × height method. These techniques are essential when perpendicular heights aren't readily available.

Formula: Area = ½ × a × b × sin C Where a and b are two sides of the triangle, and C is the included angle

This formula provides a direct method for calculating areas without needing to find the perpendicular height. It's particularly useful when working with:

  • Triangles where two sides and the included angle are known
  • Problems involving sectors and segments of circles

The relationship between this area formula and the traditional method becomes clear when you consider that sin C × b represents the perpendicular height when using side a as the base.

Example: For a triangle with sides 8cm and 10cm, and included angle 43°: Area = ½ × 8 × 10 × sin 43° = 27.3 cm²

Trigonometry
The Sine Rule
A
PROOF
b
b
Examples
A
C
с
d
1720
P
a
76
1 find the length of the side x
C
750
10
C
the sine rule
(find sides!
an

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Trigonometric Equations

Solving trigonometric equations is an essential skill in GCSE further maths. These equations involve finding values of angles that satisfy certain conditions.

Example: Solve sin θ = 0.3 for 0° ≤ θ ≤ 360°

Solution:

  1. θ = sin⁻¹(0.3) ≈ 17.5°
  2. Due to sine rule ambiguity, there's a second solution: 180° - 17.5° = 162.5°

Highlight: When solving trigonometric equations, it's often helpful to sketch the relevant graph to visualize the solutions.

The page also covers equations involving cosine and tangent functions:

  1. cos θ = -1/2 Solutions: θ = 120° or 240°

  2. tan θ = 4 Solution: θ ≈ 76.0°

Vocabulary: Theta (θ) is commonly used to represent unknown angles in trigonometric equations.

Trigonometry
The Sine Rule
A
PROOF
b
b
Examples
A
C
с
d
1720
P
a
76
1 find the length of the side x
C
750
10
C
the sine rule
(find sides!
an

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

Advanced Trigonometric Problems

This section deals with more complex trigonometric equations and introduces trigonometric identities, which are crucial for GCSE further maths and beyond.

Example: Solve 6 sin θ + 5 = 0 for 0° ≤ θ ≤ 360°

Solution:

  1. 6 sin θ = -5
  2. sin θ = -5/6
  3. θ = sin⁻¹(-5/6) ≈ -56.4°
  4. Adjusting for the given range: 303.6°
  5. Second solution: 180° - 56.4° = 123.6°

Highlight: When solving more complex equations, breaking them down into simpler steps and using known trigonometric identities can be very helpful.

The page also touches on trigonometric identities, which are equations that are true for all values of the variables involved. These identities are fundamental in simplifying and solving advanced trigonometric problems.

Vocabulary: Identities - equations that are true for all values of the variables involved, regardless of what those values may be.

This comprehensive guide covers essential topics in trigonometry for scalene triangles, providing students with the tools they need to tackle a wide range of problems in GCSE maths and beyond.

Trigonometry
The Sine Rule
A
PROOF
b
b
Examples
A
C
с
d
1720
P
a
76
1 find the length of the side x
C
750
10
C
the sine rule
(find sides!
an

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

The Sine Rule

The sine rule is a fundamental concept in trigonometry for scalene triangles. It provides a method for finding unknown sides or angles in non-right-angled triangles.

Definition: The sine rule states that the ratio of the sine of an angle to the length of the opposite side is constant for all angles in a triangle.

The formula is expressed as:

a / sin A = b / sin B = c / sin C

Where a, b, and c are the lengths of the sides, and A, B, and C are the angles opposite to those sides respectively.

Example: To find the length of side x in a triangle with angle A = 75°, side b = 10, and angle B = 72°, we use the sine rule: x / sin 75° = 10 / sin 72° x = (10 * sin 75°) / sin 72° ≈ 10.3

Highlight: When using the sine rule to find an angle, there may be two possible solutions due to the ambiguity of the inverse sine function. Always check if both solutions are valid in the context of the problem.

Vocabulary: SOHCAHTOA is a mnemonic device used to remember trigonometric ratios in right-angled triangles, but the sine rule extends these concepts to non-right-angled triangles.

Trigonometry
The Sine Rule
A
PROOF
b
b
Examples
A
C
с
d
1720
P
a
76
1 find the length of the side x
C
750
10
C
the sine rule
(find sides!
an

Sign up to see the contentIt's free!

Access to all documents

Improve your grades

Join milions of students

By signing up you accept Terms of Service and Privacy Policy

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This app is really great. There are so many study notes and help [...]. My problem subject is French, for example, and the app has so many options for help. Thanks to this app, I have improved my French. I would recommend it to anyone.

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Wow, I am really amazed. I just tried the app because I've seen it advertised many times and was absolutely stunned. This app is THE HELP you want for school and above all, it offers so many things, such as workouts and fact sheets, which have been VERY helpful to me personally.

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Thomas R

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Just amazing. Let's me revise 10x better, this app is a quick 10/10. I highly recommend it to anyone. I can watch and search for notes. I can save them in the subject folder. I can revise it any time when I come back. If you haven't tried this app, you're really missing out.

Basil

Android user

This app has made me feel so much more confident in my exam prep, not only through boosting my own self confidence through the features that allow you to connect with others and feel less alone, but also through the way the app itself is centred around making you feel better. It is easy to navigate, fun to use, and helpful to anyone struggling in absolutely any way.

David K

iOS user

The app's just great! All I have to do is enter the topic in the search bar and I get the response real fast. I don't have to watch 10 YouTube videos to understand something, so I'm saving my time. Highly recommended!

Sudenaz Ocak

Android user

In school I was really bad at maths but thanks to the app, I am doing better now. I am so grateful that you made the app.

Greenlight Bonnie

Android user

very reliable app to help and grow your ideas of Maths, English and other related topics in your works. please use this app if your struggling in areas, this app is key for that. wish I'd of done a review before. and it's also free so don't worry about that.

Rohan U

Android user

I know a lot of apps use fake accounts to boost their reviews but this app deserves it all. Originally I was getting 4 in my English exams and this time I got a grade 7. I didn’t even know about this app three days until the exam and it has helped A LOT. Please actually trust me and use it as I’m sure you too will see developments.

Xander S

iOS user

THE QUIZES AND FLASHCARDS ARE SO USEFUL AND I LOVE THE SCHOOLGPT. IT ALSO IS LITREALLY LIKE CHATGPT BUT SMARTER!! HELPED ME WITH MY MASCARA PROBLEMS TOO!! AS WELL AS MY REAL SUBJECTS ! DUHHH 😍😁😲🤑💗✨🎀😮

Elisha

iOS user

This apps acc the goat. I find revision so boring but this app makes it so easy to organize it all and then you can ask the freeeee ai to test yourself so good and you can easily upload your own stuff. highly recommend as someone taking mocks now

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