Trigonometry: Calculate Trigonometric Functions for Right Triangles

Published on June 7, 2025

Trigonometry

Find the values of sine, cosine, tangent, cosecant, secant, and cotangent for the given right triangles. The solution involves applying trigonometric ratios and the Pythagorean theorem.

Question

Answer

The user wants to find the values of the six trigonometric functions for each given right-angled triangle.

Let’s break down the problem for each triangle. The six trigonometric functions are sine (

), cosine (

), tangent (

), cosecant (

), secant (

), and cotangent (

For a right-angled triangle with an angle

We need to identify the opposite side, adjacent side, and hypotenuse for each triangle relative to the angle

Triangle A:

Opposite side = 4
Adjacent side = 3
Hypotenuse = 5

Now, calculate the 6 trigonometric functions:

Triangle B:

Opposite side = 8
Adjacent side = 15
Hypotenuse = 17

Now, calculate the 6 trigonometric functions:

Triangle C:

Opposite side = 2
Hypotenuse = 5
We need to find the adjacent side using the Pythagorean theorem: .
Let the adjacent side be .
Adjacent side =

Now, calculate the 6 trigonometric functions:

(rationalized)
(rationalized)

Triangle D:

Opposite side =
Adjacent side =
Hypotenuse =
We need to find the adjacent side. The image shows the angle at the top vertex.
Relative to this :
Opposite side =
Adjacent side =
Hypotenuse =