Cosine Rule for Side Lengths

This page introduces the cosine rule and its application in finding side lengths of non-right-angled triangles. It explains when to use the cosine rule instead of the sine rule.

Definition: The cosine rule is expressed as a² = b² + c² - 2bc cos A, where a, b, c are sides and A is the angle opposite side a.

The page provides guidance on when to use the cosine rule, specifically when given two sides and the included angle of a triangle.

Example: A triangle with sides 7cm and 8cm and an included angle of 25° is used to demonstrate the application of the cosine rule to find the third side.

This section clearly explains when to use cosine rule, which is crucial for students learning to differentiate between various trigonometric problem-solving methods.

Trigonometry: Area of a Triangle

This page introduces the formula for calculating the area of a triangle using trigonometry. The formula A = ½ab sin C is presented, where 'a' and 'b' are the lengths of two sides, and 'C' is the angle between them.

Definition: The area of a triangle can be calculated using the formula A = ½ab sin C, where 'a' and 'b' are side lengths and 'C' is the included angle.

An example problem is provided to illustrate the application of this formula. In this case, a triangle with sides 100m and 120cm and an included angle of 65° is used.

Example: For a triangle with sides 100m and 120cm and an angle of 65°, the area is calculated as A = ½ × 100 × 120 × sin 65° = 5437.846 cm².

This page serves as an introduction to trigonometry area of a triangle example problems, providing a clear foundation for students to understand and apply the concept.

Sine Rule for Finding Angles

This page focuses on using the sine rule to find unknown angles in a triangle. It presents a problem where two sides and one angle of a triangle are known, and the task is to find another angle.

Example: In a triangle with sides 8.5cm and 12cm, and an angle of 42°, the sine rule is used to find another angle.

The solution process is shown step-by-step, demonstrating how to rearrange the sine rule formula to solve for an unknown angle.

Highlight: This example illustrates the versatility of the sine rule in solving for both sides and angles in non-right-angled triangles.

This page serves as an excellent resource for students looking for trigonometry sine rule explained examples, particularly useful for GCSE and higher-level studies.

Transformations of Quadratic Functions

This page explores how different forms of quadratic functions affect their graphical representations. It covers vertical and horizontal translations of quadratic graphs.

Definition: The graph of f(x) = x² + a moves the parabola up or down the y-axis, while f(x) = (x + a)² moves the function horizontally along the x-axis.

Several examples of transformed quadratic functions are provided, along with their graphical representations.

Example: Graphs of y = x² + 3, y = x² - 2, and y = (x + 3)² are shown to illustrate different transformations.

Highlight: Understanding these transformations is key to analyzing and sketching more complex quadratic functions.

This page is particularly useful for students studying cosine rule and quadratic functions guide materials, as it connects algebraic manipulations with graphical representations.