AC9M9SP01Year 9SpacePractised on site

AC9M9SP01 — Year 9 Space

Official Curriculum Content Description:

recognise the constancy of the sine, cosine and tangent ratios for a given angle in right-angled triangles using properties of similarity

Worked Examples (2)

Example 1Trigonometric Ratios (Sine, Cosine, Tangent)

In a right-angled triangle, angle

A = 20^\circ

and the hypotenuse is

14\text{ cm}

. Find the length of the side adjacent to angle

A

x

. Give your answer to 1 decimal place.

Diagram not accurately drawn

Common mistake: Fixing opposite/adjacent to the triangle orientation rather than angle

A

, erroneously using

\sin(20^\circ)

instead of

\cos(20^\circ)

Correct Answer:

13.2\text{ cm}

Accepted: A unit is optional (cm accepted).

Worked Solution:

1. Identify the ratio:

\cos A = \frac{\text{adjacent}}{\text{hypotenuse}}

. 2. Rearrange for the adjacent side:

x = \text{hypotenuse} \times \cos A = 14 \times \cos(20^\circ)

. 3.

x \approx 13.2\text{ cm}

Example 2Inverse Trigonometric Functions

In a right-angled triangle, the side adjacent to angle

x

5\text{ cm}

and the hypotenuse is

9\text{ cm}

. Find the angle

x

in degrees. Give your answer to 1 decimal place.

Diagram not accurately drawn

Common mistake: Forgetting to apply the inverse function

\cos^{-1}

when finding an angle from a ratio, leaving the side ratio as the answer.

Correct Answer:

56.3

Worked Solution:

1. Use the inverse cosine function:

\cos x = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{5}{9}

. 2.

x = \cos^{-1}\left(\frac{5}{9}\right) = \cos^{-1}(0.5556)

. 3.

x \approx 56.3^\circ

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