Question

Use a half-angle identity to find the exact value of tan 157.5°.

Answer 1

Use the half angle identity, tan157.5=(1-cos315)/sin315
cos315=cos(-45)=√2/2, sin315=-√2/2
tan157.5=(1-√2/2)/(-√2/2)=(2-√2)/(-√2)=(2√2-2)/(-2)=1-√2

Answer 2

Answer:

$tan157.5^{\circ}=-0.414$

Step-by-step explanation:

We are given that $tan157.5^{\circ}$

We have to find the exact value of $tan157.5^{\circ}$ by using a half -angle identity.

We know that

$tan\frac{\theta}{2}=\frac{1-cos\theta}{sin\theta}$

Therefore, we have

$\frac{\theta}{2}=157.5^{\circ}$

$\theta=157.5\times 2=315^{\circ}$

Now, by using half angle identity

$\tan157.5^{\circ}=\frac{1-cos315^{\circ}}{sin315^{\circ}}$

$\tan157.5^{\circ}=\frac{1-0.707}{-0.707}$

$tan157.5^{\circ}=-0.414$