Checking and securing understanding of cosine problems | Maths

Starter quiz

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

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

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

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

$43^\circ$ ✓

$47^\circ$

$90^\circ$
$What is the approximate value of <Math>\cos(60^\circ)</Math>?$

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

'0.5' ✓
$What is the approximate value of <Math>\cos(30^\circ)</Math>?$

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

'0.87' ✓

Exit quiz

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

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

'10 cm' ✓
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?

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

$\theta=\arccos\left(\frac{\text{adj}}{\text{hyp}}\right)$ ✓

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

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

$\text{hyp}=\frac{\text{adj}}{\cos(\theta)}$ ✓
Which of the following equations are correct when finding the angle $n^\circ$ in the equation: $\cos(n^\circ)=0.57$ ?

$n=\cos(0.57)$

$n=\arccos(0.57)$ ✓

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

$n=\cos^{-1}(0.57)$ ✓
Which of these is correct for this triangle?

$\cos(x)=\frac{3}{5}$ ✓

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

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

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

$\cos(x)=\frac{4}{3}$
Calculate the length of $x$ to 2 decimal places.

'3.76 cm' ✓

Worksheet

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Lesson Details

Key learning points

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.