Suppose for an angle theta in a right triangle cos theta = C. Sketch and label this triangle, and then use it to write the other five trig functions of theta in terms of C.

Question

asked Jun 6, 2021 222k views

1 Answer

Sashkello · Answer 1 · 2021-06-13T05:20:00+0000

Answer:

$sin\theta = √(1-C^2)$

$tan\theta = (√(1-C^2))/(C)$

$cot\theta = (C)/(√(1-C^2))$

$sec\theta = (1)/(C)}$

$cosec\theta = (1)/(√(1-C^2))$

Explanation:

Given that:

$\theta$ is an angle in a right angled triangle.

and
$cos\theta = C$

To find:

To draw the triangle and write other five trigonometric functions in terms of C.

Solution:

We know that cosine of an angle is given by the formula:

cosx =(Base)/(Hypotenuse)

Here, we are given that
$cos\theta = C$ OR

$cos\theta = (C)/(1)$

i.e. Base = C and Hypotenuse of triangle = 1

Please refer to the right angled triangle as per given statements.

$\triangle PQR$ , with base PR = C units

and hypotenuse, QP = 1 unit

$\angle R$ is the right angle.

Let us use pythagorean theorem to find the value of perpendicular.

According to pythagorean theorem:

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

$1^(2) = C^(2) + QR^(2)\\\Rightarrow QR = \sqrt {1-C^2}$

$sin\theta = (Perpendicular)/(Hypotenuse)\\\Rightarrow sin\theta = (√(1-C^2))/(1)\\\Rightarrow sin\theta = √(1-C^2)$

$tan\theta = (Perpendicular)/(Base)\\\Rightarrow tan\theta = (√(1-C^2))/(C)$

$cot\theta = (Base)/(Perpendicular)\\\Rightarrow cot\theta = (C)/(√(1-C^2))$

$sec\theta = (Hypotenuse)/(Base)\\\Rightarrow sec\theta = (1)/(C)}$

$cosec\theta = (Hypotenuse)/(Perpendicular)\\\Rightarrow cosec\theta = (1)/(√(1-C^2))$