Question

Sin theta = 3/4, what are the values of cos theta and tan theta?

Answer 1

`\displaystyle \cos(\theta) = (√(7))/(4)`.

`\displaystyle \tan(\theta) = (3√(7))/(7)`.

Explanation:

By the Pythagorean identity, for any given angle
`\theta`,
`\sin^(2)(\theta) + \cos^(2)(\theta) = 1`.

Given that
`\displaystyle \sin(\theta) = 3/4`, solve this equation for
`\cos(\theta)`.

`(\sin(\theta))^(2) + (\cos(\theta))^(2) = 1`.

`\displaystyle \left((3)/(4)\right)^(2) + (\cos(\theta))^(2) = 1`.

`\begin{aligned} \cos(\theta) &= \sqrt{1 - (9)/(16)} \\ &= \sqrt{(7)/(16)} \\ &= (√(7))/(4) \end{aligned}`.

The tangent of an angle is equal to the ratio between the sine and cosine of that angle. In this question:

`\begin{aligned} \tan(\theta) &= (\sin(\theta))/(\cos(\theta)) \\ &= (3/4)/(√(7)/4) \\ &= (3)/(√(7)) \\ &= (3√(7))/(7)\end{aligned}`.