Question

whats the exact value of tan 15°?
a) 2 + √3
b) 1
c) 2 - √3
d) 1 + √3
question 19 (5 points)
which of the following is equal to cos θ using coordinates on the unit circle?
a) y/x
b) y
c) x

Explanation:

Step1: Express 15° as 45° - 30°

Use the tangent - difference formula $\tan(A - B)=\frac{\tan A-\tan B}{1 + \tan A\tan B}$, where $A = 45^{\circ}$ and $B=30^{\circ}$.

Step2: Substitute the values of $\tan45^{\circ}$ and $\tan30^{\circ}$

We know that $\tan45^{\circ}=1$ and $\tan30^{\circ}=\frac{\sqrt{3}}{3}$. Then $\tan(45^{\circ}-30^{\circ})=\frac{\tan45^{\circ}-\tan30^{\circ}}{1+\tan45^{\circ}\tan30^{\circ}}=\frac{1-\frac{\sqrt{3}}{3}}{1 + 1\times\frac{\sqrt{3}}{3}}$.

Step3: Simplify the expression

First, multiply the numerator and denominator by 3 to get $\frac{3-\sqrt{3}}{3 + \sqrt{3}}$. Then rationalize the denominator by multiplying the numerator and denominator by $3-\sqrt{3}$:
\[

$$\begin{align*} \frac{(3 - \sqrt{3})(3-\sqrt{3})}{(3+\sqrt{3})(3 - \sqrt{3})}&=\frac{9-6\sqrt{3}+3}{9 - 3}\\ &=\frac{12-6\sqrt{3}}{6}\\ &=2-\sqrt{3} \end{align*}$$

\]

For the second question:
On the unit - circle, if a point on the unit - circle has coordinates $(x,y)$ and the angle $\theta$ is measured counter - clockwise from the positive x - axis, then $\cos\theta=x$ and $\sin\theta = y$.

Answer:

1. C. $2-\sqrt{3}$
2. C. $x$