Question

""Express the following trigonometric expressions as algebraic expressions in terms of ‘x’:

1. sin(arccos(x))
2. tan(arccos(x))""

Answer 1

Using the right triangle approach, sin(arccos(x)) can be expressed as √(1 - x²), and tan(arccos(x)) as √(1 - x²)/x.

Step-by-step explanation:

To express the trigonometric expressions sin(arccos(x)) and tan(arccos(x)) as algebraic expressions in terms of 'x', we will use a right triangle approach.

sin(arccos(x))

Imagine a right triangle where the adjacent side to an angle A is 'x' and the hypotenuse is 1 (since the cosine of an angle is the adjacent side over the hypotenuse, arccos(x) suggests that cos(A) = x/1 = x). Therefore, the opposite side can be found using Pythagoras' theorem: opposite side = √(1 - x²). Now, sin(A) is the opposite side over the hypotenuse, which in this case is √(1 - x²)/1 = √(1 - x²). So, sin(arccos(x)) = √(1 - x²).

tan(arccos(x))

To find tan(arccos(x)), we again consider the same right triangle. We already have the opposite side and the adjacent side. Tan(A), which is the opposite side over the adjacent side, equals √(1 - x²)/x. Therefore, tan(arccos(x)) = √(1 - x²)/x.