Question

If tan x = .76 and cos x = .22, what is sin x?



If sin x = .42, what is cos x?

Answer 1

Conditions

1) If $\tan x = .76$ and $\cos x = .22$, what is $\sin x$?

2) If $\sin x = .42$, what is $\cos x$?

Solution

1)

This is a trigonometric task. Here we must use the Basic Trigonometric Identity. Its formula is:

$$\sin^2(x) + \cos^2(x) = 1, \, x \in \mathbb{R}.$$

As it known, $\tan(x) = \frac{\sin(x)}{\cos(x)}$, so

$$\frac{\sin(x)}{\cos(x)} = 0.76 \Rightarrow \sin(x) = 0.76 \cos(x). \quad \cos(x) = 0.22 \Rightarrow \sin(x) = 0.76 \times 0.22 = 0.1672$$

2)

This task is logically near previous. We use the same formula:

$$\sin^2(x) + \cos^2(x) = 1, \, x \in \mathbb{R}.$$ $$\begin{array}{l} \sin^2(x) + \cos^2(x) = (0.42)^2 + \cos^2(x) = 1 \Rightarrow \cos^2(x) = 1 - 0.1764 = 0.8236 \\ \Rightarrow \cos(x) = \pm \sqrt{0.8236} \\ \end{array}$$

This value is irrational, so we can leave it in this form.

Answer:

1) $\sin(x) = 0.1672$

2) Two solutions: $\cos(x) = \sqrt{0.8236}$ or $\cos(x) = -\sqrt{0.8236}$