Question

The value of tangent x is given. Find sine x and cos x if x lies in the specified interval.
tan x = 21​, x is an element of [0, pi / 2]

Answer 1

Answer:

sin x = 0.998

cosx = 0.046

Step-by-step explanation:

Given that:

tan x = 21

where interval of x is $[0,\dfrac{\pi}{2}]$.

We know that the trigonometric identity for tan x is:

$tan\theta = \dfrac{Perpendicular}{Base}$

Comparing with:

$tan x = \dfrac{21}{1}$

Perpendicular = 21 units

Base = 1 unit

As per pythagorean theorem:

$\text{Hypotenuse}^{2} = \text{Base}^{2} + \text{Perpendicular}^{2}$

$\Rightarrow \text{Hypotenuse}^2 = 21^2 +1^2\\\Rightarrow \text{Hypotenuse} = \sqrt{442} = 21.023\ units$

interval of x is $[0,\dfrac{\pi}{2}]$ so values of sinx and cosx will be positive because it is first quadrant where values of sine and cosine are positive.

We know that

$sin\theta = \dfrac{Perpendicular}{Hypotenuse}\\cos\theta = \dfrac{Base}{Hypotenuse}$

So, sine x :

$\Rightarrow sinx =\dfrac{21}{21.023}\\\Rightarrow sinx = 0.998$

Similarly, value of cos x :

$\Rightarrow cosx =\dfrac{1}{21.023}\\\Rightarrow cosx = 0.046$