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Kate Robinson
@katerobinson_rafs
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This document provides a comprehensive guide on trigonometry, covering the sine rule, cosine rule, and bearings. It includes detailed explanations, examples, and step-by-step solutions for various trigonometric problems.
05/07/2022
232
Understanding trigonometric graphs is crucial for solving trigonometric equations and visualizing periodic functions. The main trigonometric functions are sine, cosine, and tangent.
Key features of trigonometric graphs:
Sine graph:
Cosine graph:
Tangent graph:
Highlight: Memorizing key points on these graphs (0°, 30°, 45°, 60°, 90°) can be very helpful in sketching and interpreting them quickly.
The page also introduces the concept of sine rule ambiguity. When using the sine rule to find an angle, there may be two possible solutions due to the nature of the inverse sine function.
Example: In a triangle ABC with ∠ABC = 30°, AB = 10 cm, and AC = 7 cm, there are two possible values for ∠ACB:
Vocabulary: Obtuse angle - an angle greater than 90° but less than 180°.
The cosine rule is another essential tool in trigonometry for scalene triangles. It is particularly useful when dealing with triangles where you know three sides or two sides and the included angle.
Definition: The cosine rule states that in a triangle with sides a, b, and c, and angles A, B, and C opposite these sides respectively: a² = b² + c² - 2bc cos A
This formula can be rearranged to find an angle:
cos A = (b² + c² - a²) / (2bc)
Example: To find the length of side x in a triangle with sides 4 and 3, and an included angle of 80°: x² = 4² + 3² - 2 * 4 * 3 * cos 80° x² ≈ 20.822 x ≈ 4.56 (to 3 significant figures)
Highlight: The cosine rule is a generalization of the Pythagorean theorem for non-right-angled triangles.
Vocabulary: Sig figs (significant figures) are important in presenting accurate results in trigonometric calculations.
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:
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.
Calculating the area of a non-right angle triangle is an important application of trigonometry. There are several methods to find the area, depending on the given information.
Formula: The area of a triangle can be calculated using the formula: Area = ½ * a * b * sin C
Where a and b are the lengths of two sides, and C is the included angle.
Example: For a triangle with sides 10 cm and 8 cm, and an included angle of 43°: Area = ½ * 10 * 8 * sin 43° ≈ 27.3 cm² (to 3 significant figures)
Highlight: This formula is particularly useful when you don't have a perpendicular height available, which is often the case in scalene triangles.
The page also covers the area of a segment, which is the region of a circle enclosed by an arc and a chord. This involves subtracting the area of a triangle from the area of a sector.
Example: For a segment with radius 10.4 cm and central angle 120°:
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:
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:
cos θ = -1/2 Solutions: θ = 120° or 240°
tan θ = 4 Solution: θ ≈ 76.0°
Vocabulary: Theta (θ) is commonly used to represent unknown angles in trigonometric equations.
Applying trigonometry to bearings problems is a practical application of trigonometry for scalene triangles. Bearings are used in navigation and require a specific approach to angle measurement.
Key points for working with bearings:
Example: A ship A is 10 km from a lighthouse L on a bearing of 115°. Another ship B is 14.5 km from L on a bearing of 200°. To find the distance between the ships:
Highlight: When solving bearing problems, always start by drawing a clear diagram with north clearly marked.
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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Kate Robinson
@katerobinson_rafs
·
33 Followers
Follow
This document provides a comprehensive guide on trigonometry, covering the sine rule, cosine rule, and bearings. It includes detailed explanations, examples, and step-by-step solutions for various trigonometric problems.
Understanding trigonometric graphs is crucial for solving trigonometric equations and visualizing periodic functions. The main trigonometric functions are sine, cosine, and tangent.
Key features of trigonometric graphs:
Sine graph:
Cosine graph:
Tangent graph:
Highlight: Memorizing key points on these graphs (0°, 30°, 45°, 60°, 90°) can be very helpful in sketching and interpreting them quickly.
The page also introduces the concept of sine rule ambiguity. When using the sine rule to find an angle, there may be two possible solutions due to the nature of the inverse sine function.
Example: In a triangle ABC with ∠ABC = 30°, AB = 10 cm, and AC = 7 cm, there are two possible values for ∠ACB:
Vocabulary: Obtuse angle - an angle greater than 90° but less than 180°.
The cosine rule is another essential tool in trigonometry for scalene triangles. It is particularly useful when dealing with triangles where you know three sides or two sides and the included angle.
Definition: The cosine rule states that in a triangle with sides a, b, and c, and angles A, B, and C opposite these sides respectively: a² = b² + c² - 2bc cos A
This formula can be rearranged to find an angle:
cos A = (b² + c² - a²) / (2bc)
Example: To find the length of side x in a triangle with sides 4 and 3, and an included angle of 80°: x² = 4² + 3² - 2 * 4 * 3 * cos 80° x² ≈ 20.822 x ≈ 4.56 (to 3 significant figures)
Highlight: The cosine rule is a generalization of the Pythagorean theorem for non-right-angled triangles.
Vocabulary: Sig figs (significant figures) are important in presenting accurate results in trigonometric calculations.
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:
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.
Calculating the area of a non-right angle triangle is an important application of trigonometry. There are several methods to find the area, depending on the given information.
Formula: The area of a triangle can be calculated using the formula: Area = ½ * a * b * sin C
Where a and b are the lengths of two sides, and C is the included angle.
Example: For a triangle with sides 10 cm and 8 cm, and an included angle of 43°: Area = ½ * 10 * 8 * sin 43° ≈ 27.3 cm² (to 3 significant figures)
Highlight: This formula is particularly useful when you don't have a perpendicular height available, which is often the case in scalene triangles.
The page also covers the area of a segment, which is the region of a circle enclosed by an arc and a chord. This involves subtracting the area of a triangle from the area of a sector.
Example: For a segment with radius 10.4 cm and central angle 120°:
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:
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:
cos θ = -1/2 Solutions: θ = 120° or 240°
tan θ = 4 Solution: θ ≈ 76.0°
Vocabulary: Theta (θ) is commonly used to represent unknown angles in trigonometric equations.
Applying trigonometry to bearings problems is a practical application of trigonometry for scalene triangles. Bearings are used in navigation and require a specific approach to angle measurement.
Key points for working with bearings:
Example: A ship A is 10 km from a lighthouse L on a bearing of 115°. Another ship B is 14.5 km from L on a bearing of 200°. To find the distance between the ships:
Highlight: When solving bearing problems, always start by drawing a clear diagram with north clearly marked.
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.
Maths - Symmetry GCSE
Includes different lines of symmetry and exam style questions
4
185
0
Maths - Enlargement and describing enlargements
Learn how to calculate the scale factor for enlarging shapes and count the number of squares in the new shape after enlargement.
3
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1
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Column Vectors and transformation flow chart
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Average app rating
Pupils love Knowunity
In education app charts in 12 countries
Students have uploaded notes
iOS User
Philip, iOS User
Lena, iOS user
