Question

Use a Pythagorean identity to find sin-theta if cos-theta=3/7 and the terminal side of lies in quadrant IV. Rationalize the denominator, if necessary. Give exact answer. sin-theta= ?

Answer 1

Conditions

Use a Pythagorean identity to find $\sin$-theta if $\cos$-theta = 3/7 and the terminal side of lies in quadrant IV. Rationalize the denominator, if necessary. Give exact answer. $\sin$-theta = ?

Solution

Pythagorean identity:

$$\sin^2 \theta + \cos^2 \theta = 1$$ $$\sin^2 \theta + \frac{9}{49} = 1$$ $$\sin^2 \theta = \frac{40}{49}$$ $$\sin \theta = \pm \frac{2\sqrt{10}}{7}$$

As we know, that the terminal side lies in quadrant IV, then:

$$\sin \theta < 0$$ $$\theta = -\arcsin \left( \frac{2\sqrt{10}}{7} \right) + 2\pi n, \text{ n∈Z}$$