Question

Write the value of \[sin\left( {25 - a} \right) - cos\left( {65 + a} \right)\] .

Answer 1

Hint: The trigonometric ratios of a triangle are also called the trigonometric functions. Sine, cosine, and tangent are 3 important trigonometric functions and are abbreviated as sin, cos and tan. Here, we are given a trigonometric function expression. We need to find the value of it. We know that \[\sin (90 - \theta ) = \cos \theta \] and so applying this, we will get the final output.

Complete step-by-step answer:
Trigonometry, as the name might suggest, is all about triangles. The six essential trigonometric functions are sine, cosine, secant, cosecant, tangent, and cotangent. Out of them, the main functions in trigonometry are Sine, Cosine and Tangent. They are simply one side of a right-angled triangle divided by another. The triangle could be larger, smaller or turned around, but that angle will always have that ratio.
Given that,
\[sin\left( {25 - a} \right) - cos\left( {65 + a} \right)\]
We will use 90 – 65 = 25, and use this in sin function, we will get,
\[ = \sin [(90 - 65) - a] - \cos (65 + a)\]
Removing the brackets, we will get,
\[ = \sin [90 - 65 - a] - \cos (65 + a)\]
\[ = \sin [90 - (65 + a)] - \cos (65 + a)\]
We know that, \[\sin (90 - \theta ) = \cos \theta \], and so applying this, we will get,
\[ = \cos (65 + a) - \cos (65 + a)\]
\[ = 0\]
Hence, the value of \[sin\left( {25 - a} \right) - cos\left( {65 + a} \right) = 0\]
So, the correct answer is “0”.

Note: Trigonometry is one of those divisions in mathematics that helps in finding the angles and missing sides of a triangle with the help of trigonometric ratios. The angles are either measured in radians or degrees. The concept of unit circle helps us to measure the angles of cos, sin and tan directly since the centre of the circle is located at the origin and radius is 1. When learning about trigonometric formulas, we need to consider only right-angled triangles. However, they can be applied to other triangles also.