The cosine ratio involves the hypotenuse, adjacent and the angle
If you know the length of the hypotenuse and the size of the angle, you can use the cosine ratio
If you know the length of the adjacent and the size of the angle, you can use the cosine ratio
If you know the length of the hypotenuse and adjacent, you can use the cosine ratio

Common misconception

The cosine formula is only used to find the length of a side adjacent to an angle.

The cosine formula can be used to find the length of a side adjacent to an angle. A rearrangement of the formula also allows us to find the length of the hypotenuse given the adjacent side. The arccosine function allows us to find the angle, itself.

Keywords

Hypotenuse - The hypotenuse is the side of a right-angled triangle which is opposite the right angle.
Adjacent - The adjacent side of a right-angled triangle is the side which is next to both the right angle and the marked angle.
Trigonometric functions - Trigonometric functions are commonly defined as ratios of two sides of a right-angled triangle containing the angle.
Cosine function - The cosine of an angle (cos(θ°)) is the x-coordinate of point P on the triangle formed inside the unit circle.

It may be helpful to introduce the cosine formula as h × cos(θ°) = adj, that is to say the length of the side adjacent to an angle is equal to the length of the hypotenuse multiplied by the cosine of that angle, so that both rearrangements of the formula can be shown using a one-step division.

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This content is © Oak National Academy Limited (2024), licensed on Open Government Licence version 3.0 except where otherwise stated. See Oak's terms & conditions (Collection 2).

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Starter quiz

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6 Questions

Q1.

Two shapes are similar if the only difference between them is their . Their side lengths are in the same proportions.

Correct Answer: size

Q2.

Triangle ABC and triangle DEF are similar. What is the length of side $$x$$?

Correct Answer: 40 cm, forty, 40

Q3.

Triangle ABC and triangle DEF are similar. What is the length of side $$x$$?

Correct Answer: 10 cm, 10, ten

Q4.

The two triangles are similar. What is the size of the angle marked $$x$$?

Correct answer: $$43^\circ$$

$$47^\circ$$

$$90^\circ$$

Q5.

What is the approximate value of $$\cos(60^\circ)$$?

Correct Answer: 0.5

Q6.

What is the approximate value of $$\cos(30^\circ)$$?

Correct Answer: 0.87

Exit quiz

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6 Questions

Q1.

Work out the length of $$x$$, given that these two triangles are similar.

Correct Answer: 4.74 cm, 4.74

Q2.

Work out the length of $$x$$, given that these two triangles are similar.

Correct Answer: 10 cm, 10, ten

Q3.

For a right-angled triangle, the cosine ratio is $$\cos(\theta)=\frac{\text{adj}}{\text{hyp}}$$, where $$\theta$$ is the angle in degrees. Which of these are equivalent forms?

$$\text{hyp}=\frac{\cos(\theta)}{\text{adj}}$$

Correct answer: $$\theta=\arccos\left(\frac{\text{adj}}{\text{hyp}}\right)$$

Correct answer: $$\text{hyp}\times\cos(\theta)=\text{adj}$$

$$\cos(\theta)=\frac{\text{hyp}}{\text{adj}}$$

Correct answer: $$\text{hyp}=\frac{\text{adj}}{\cos(\theta)}$$

Q4.

Which of the following equations are correct when finding the angle $$n^\circ$$ in the equation: $$\cos(n^\circ)=0.57$$?

$$n=\cos(0.57)$$

Correct answer: $$n=\arccos(0.57)$$

$$n=\frac{0.57}{\cos()}$$

Correct answer: $$n=\cos^{-1}(0.57)$$

Q5.

Which of these is correct for this triangle?

Correct answer: $$\cos(x)=\frac{3}{5}$$

$$\cos(x)=\frac{3}{4}$$

$$\cos(x)=\frac{4}{5}$$

$$\cos(x)=\frac{5}{3}$$

$$\cos(x)=\frac{4}{3}$$

Q6.

Calculate the length of $$x$$ to 2 decimal places.

Correct Answer: 3.76 cm, 3.76