NZ Level 7 (NZC) Level 2 (NCEA)
Tangent Ratios (proofs and equivalences)

## Interactive practice questions

By finding the ratio represented by $\sin\theta$sinθ, $\cos\theta$cosθ and $\tan\theta$tanθ in the given figure, we want to prove that $\frac{\sin\theta}{\cos\theta}=\tan\theta$sinθcosθ=tanθ.

a

Write down the expression for $\sin\theta$sinθ.

b

Write down the expression for $\cos\theta$cosθ.

c

Hence, form an expression for $\frac{\sin\theta}{\cos\theta}$sinθcosθ.

d

Write down the expression for $\tan\theta$tanθ.

e

Does $\frac{\sin\theta}{\cos\theta}=\tan\theta$sinθcosθ=tanθ?

Yes

A

No

B

Yes

A

No

B
Easy
Approx 2 minutes

Which of the following options correctly completes the identity?

$\tan z=\editable{}$tanz=

Given that $\sin x=\frac{4}{5}$sinx=45 and $\cos x=\frac{3}{5}$cosx=35, find $\tan x$tanx.

Use properties of the trigonometric functions to find the exact value of $\tan10^\circ-\frac{\sin10^\circ}{\cos10^\circ}$tan10°sin10°cos10°

### Outcomes

#### M7-4

Apply trigonometric relationships, including the sine and cosine rules, in two and three dimensions