Question

Write the expression in algebraic form. \(\cos \left(\arcsin \frac{x-h}{r}\right)\)

Short answer

\(\cos \left(\arcsin \frac{x-h}{r}\right) = \frac{\sqrt{r^2 - (x - h)^2}}{r}\)

Step-by-step solution

Recognize the relation between sine and cosine

Recall that the sine and cosine of an angle are related according to the Pythagorean identity. It is given by \(\cos^2 \theta + \sin^2 \theta = 1\). Therefore, \(\cos \theta = \sqrt{1 - \sin^2 \theta}\). Note that the sign of cosine depends on the quadrant in which \(\theta\) lies.

Substitute arcsin expression into identity

Next replace sin term in the above identity with the provided information. We are given \(\sin \theta = \frac{x - h}{r}\). Using the relation from step 1, we can substitute this into the identity to get: \(\cos \theta = \sqrt{1 - \left(\frac{x - h}{r}\right)^2}\).

Simplify

Now, simplify the expression inside the square root to make the expression neater. This results in: \(\cos \theta = \sqrt{1 - \frac{(x - h)^2}{r^2}} = \frac{\sqrt{r^2 - (x - h)^2}}{r}\).

Key concepts

Pythagorean Identity

To understand the algebraic expression of trigonometric functions, one of the foundational concepts we start with is the Pythagorean identity. This identity is a central pillar in trigonometry and is derived from the Pythagorean theorem, which relates the sides of a right triangle.

The identity states that for any angle \(\theta\) in a right triangle, the square of the sine of \(\theta\), plus the square of the cosine of \(\theta\), always equals one: \[\cos^2 \theta + \sin^2 \theta = 1.\] This elegant relationship allows us to find the value of one trigonometric function if we know the value of the other. It is crucial for understanding the interplay between different trigonometric functions and for simplifying expressions involving trigonometric functions.

When we apply this identity, as seen in the exercise, we can solve for one trigonometric function (cosine) in terms of another (sine), given specific values or expressions. Moreover, it can help us determine the sign of the trigonometric functions based on the quadrant in which the angle \(\theta\) resides.

Sine and Cosine Relationship

The sine and cosine functions are two of the primary trigonometric functions and they have a very special relationship. This connection is highlighted through the Pythagorean identity, but more fundamentally, sine and cosine are defined based on positions on the unit circle or ratios in a right triangle.

For any acute angle \(\theta\), the sine function \(\sin(\theta)\) represents the ratio of the length of the opposite side to the hypotenuse. Conversely, the cosine function \(\cos(\theta)\) represents the ratio of the adjacent side to the hypotenuse: \[\sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}},\quad \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}}.\]

In solving our exercise, recognizing the relationship between these two functions was crucial. We noted that solving for \(\cos(\theta)\) when given \(\sin(\theta)\) is possible because of their inherent connection through their definitions and the Pythagorean identity. The algebraic manipulation of this identity allowed us to express cosine in terms of sine.

Inverse Trigonometric Functions

Inverse trigonometric functions, also known as arcfunctions, are the reverse processes of the trigonometric functions. Where trigonometric functions give you the ratio based on an angle, inverse trigonometric functions provide you with an angle given a ratio.

For instance, the arcsine function \(\arcsin(x)\) will give you an angle whose sine equals \(x\). Similarly, the function \(\arccos(x)\) will give you an angle whose cosine equals \(x\). These functions are key when we need to derive an angle from a trigonometric value.

In our exercise, we encountered the \(\arcsin\) function, represented as \(\arcsin\left(\frac{x-h}{r}\right)\). This meant we were looking for an angle \(\theta\) whose sine equalled \(\frac{x-h}{r}\). Once we have this angle \(\theta\), as postulated by the inverse function, we can then find the corresponding cosine value. This approach of moving from a ratio to an angle and then to another trigonometric function is a powerful tool for solving many kinds of trigonometric problems.