Question

Simplify cos \(\left(90^{\circ}-x\right)\) using a difference identity.

Short answer

cos(90^{\circ} - x) = sin(x)

Step-by-step solution

Identify the Trigonometric Identity

The given expression is \(\text{cos}(90^{\circ} - x)\). To simplify this, recognize that it can be addressed using the co-function identity for cosine and sine: \(\text{cos}(90^{\circ} - x) = \text{sin}(x)\).

Apply the Co-Function Identity

Using the co-function identity for cosine and sine, substitute directly to get: \(\text{cos}(90^{\circ} - x) = \text{sin}(x)\).

Key concepts

Cosine

Cosine, often abbreviated as 'cos', is a fundamental trigonometric function. It relates the angle of a right triangle to the lengths of the adjacent side and hypotenuse. If you have a right triangle with an angle \( \theta \) , the cosine of \( \theta \) is the ratio of the length of the adjacent side to the length of the hypotenuse. Mathematically, this can be written as: \[ \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \] This function is essential in many areas of mathematics, physics, and engineering.

Sine

Sine, abbreviated as 'sin', is another critical trigonometric function. It also relates the angle of a right triangle to side lengths, but focuses on the opposite side and hypotenuse. Given a right triangle with an angle \( \theta \) , the sine of \( \theta \) is the ratio of the opposite side's length to the hypotenuse's length: \[ \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \] Like cosine, sine is widely used in various scientific and engineering fields.

Co-Function Identities

Co-function identities are special relationships between trigonometric functions. One of the most useful co-function identities is the relationship between sine and cosine. Specifically, for any angle \( x \), the co-function identities state that: \[ \cos(90^{\circ} - x) = \sin(x) \] \[ \sin(90^{\circ} - x) = \cos(x) \] These identities are helpful when you need to simplify expressions or solve trigonometric equations, as they allow you to convert between sine and cosine functions easily.

Simplifying Trigonometric Expressions

Simplifying trigonometric expressions often involves using identities and relationships between the functions. In the given exercise, to simplify \( \cos(90^{\circ} - x) \), we used the co-function identity \( \cos(90^{\circ} - x) = \sin(x) \). Here are some general steps for simplifying such expressions:

• Identify which trigonometric identity or co-function identity could be applied.
• Substitute the identity into the original expression.
• Simplify the expression further, if possible.

This approach helps break down complex expressions into simpler terms, making them easier to work with.