Question

Right-triangle relationships Draw a right triangle to simplify the given expressions. Assume \(x>0.\) $$\cos \left(\sin ^{-1} x\right)$$

Short answer

In the given expression, we need to simplify \(\cos(\sin^{-1}(x))\). To do this, we first draw a right triangle and label one of the angles as angle \(A\), the side opposite of angle \(A\) as \(x\), the hypotenuse as \(H\), and the side adjacent to angle \(A\) as \(A_C\). Knowing that \(\sin A = \frac{x}{H}\), we then use the Pythagorean theorem to find the length of side \(A_C\), which is \(\sqrt{H^2 - x^2}\). Finally, we calculate the cosine function of angle \(A\) using the adjacent side and the hypotenuse, which gives us \(\cos(\sin^{-1}(x)) = \frac{\sqrt{H^2 - x^2}}{H}\).

Step-by-step solution

Create the right triangle

Draw a right triangle, and label one of the acute angles as angle \(A\). Let the side opposite to angle \(A\) be \(x\). Let the hypotenuse be \(H\). The other side, adjacent to angle \(A\), will be labeled as \(A_C\) (Adjacent to Angle \(A\)).

Identify the given information

In the expression, we are given the value of \(\sin^{-1}(x)\). This means that the sine function of angle \(A\) is \(x\). In a right triangle, we know that: $$\sin A = \frac{\text{opposite}}{\text{hypotenuse}}$$ So, this means: $$\sin A = \frac{x}{H}$$

Find the adjacent side

In order to find \(\cos(\sin^{-1}(x))\), which is the cosine function of angle \(A\), we need to find the adjacent side \(A_C\). Using the Pythagorean theorem, we can find the length of side \(A_C\): $$A_C^2 + x^2 = H^2$$ Solve for \(A_C\): $$A_C = \sqrt{H^2 - x^2}$$

Calculate the cosine function of angle \(A\)

Using the right triangle, the cosine function of angle \(A\) is: $$\cos A = \frac{\text{adjacent}}{\text{hypotenuse}}$$ Substitute the values of \(A_C\) and \(H\): $$\cos A = \frac{\sqrt{H^2 - x^2}}{H}$$ Since angle \(A\) is by definition equal to \(\sin^{-1}(x)\), the given expression becomes: $$\cos \left(\sin ^{-1} x\right) = \frac{\sqrt{H^2 - x^2}}{H}$$

Key concepts

Right Triangle

A right triangle is one of the most important shapes in trigonometry, featuring a right angle (90 degrees). It is the foundation for defining the trigonometric ratios—sine, cosine, and tangent. When working with right triangles, you'll often encounter problems that refer to the lengths of the sides relative to a specific angle within the triangle. These sides are known as the opposite, the adjacent, and the hypotenuse. The hypotenuse is the side across from the right angle and is also the longest side in the right triangle. Understanding the relationships between these sides is essential for solving various trigonometric problems.

When given a problem like \(\cos (\sin^{-1} x)\), visualizing and drawing a right triangle can greatly help in simplifying and understanding the expression. By establishing a physical model of the triangle, we enable a straightforward application of trigonometric identities and theorems.

Sine and Cosine

Sine and cosine are fundamental trigonometric functions that associate an angle of a right triangle with ratios of two of its sides. The sine of an angle is the ratio of the length of the opposite side to the length of the hypotenuse. Conversely, the cosine of an angle is the ratio of the length of the adjacent side to the hypotenuse.

The identities and values of sine and cosine functions are critical for solving problems involving right triangles. In the given exercise, the inverse sine function (also known as arcsine) is used, which corresponds to finding an angle whose sine is x. Knowing that sine relates to the opposite side and hypotenuse, and cosine relates to the adjacent side and hypotenuse, allows us to connect the given angles and side lengths through their trigonometric functions.

Pythagorean Theorem

A keystone in the study of geometry, the Pythagorean theorem is a mathematical principle that relates the lengths of the sides in a right-angled triangle. The theorem states that the square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides. In a formula, it is expressed as:
\[a^2 + b^2 = c^2\]
where \(a\) and \(b\) are the lengths of the legs, and \(c\) is the length of the hypotenuse. When dealing with inverse trigonometric functions, the Pythagorean theorem helps us find unknown side lengths. In the exercise, we used it to find the length of the adjacent side after we were given the length of the opposite side (from the sine function) and the hypotenuse.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the variables they contain. They are indispensable tools for simplifying expressions, solving equations, and transforming the form of trigonometric expressions. One of the most basic and widely used identities is the Pythagorean identity, which relates the sine and cosine of the same angle:
\[\sin^2 A + \cos^2 A = 1\]
This is derived from the Pythagorean theorem and confirms the intrinsic link between the geometrical understanding of a right triangle and its trigonometric properties. Our exercise showcased how to use identities and inverse functions together to evaluate \(\cos (\sin^{-1} x)\) by interpreting the inverse sine as an angle in a right triangle and subsequently using the Pythagorean identity to solve for the adjacent side.