Question

Find each value without using a calculator $$ \cos \left[\cos ^{-1}\left(\frac{4}{5}\right)+\sin ^{-1}\left(\frac{12}{13}\right)\right] $$

Short answer

\(-\frac{16}{65}\)

Step-by-step solution

Understanding the Inverse Functions

We start by understanding that the expression involves inverse trigonometric functions. Specifically, \( \cos^{-1} \left(\frac{4}{5}\right) \) represents an angle whose cosine is \( \frac{4}{5} \), and \( \sin^{-1} \left(\frac{12}{13}\right) \) represents an angle whose sine is \( \frac{12}{13} \). Let's denote these angles by \( \theta \) and \( \phi \), respectively.

Finding Sine of Angle \(\theta\)

Since \( \cos(\theta) = \frac{4}{5} \), we can find \( \sin(\theta) \) using the identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \). Thus, \( \sin^2(\theta) = 1 - \left(\frac{4}{5}\right)^2 = 1 - \frac{16}{25} = \frac{9}{25} \). Therefore, \( \sin(\theta) = \frac{3}{5} \).

Finding Cosine of Angle \(\phi\)

Since \( \sin(\phi) = \frac{12}{13} \), we can find \( \cos(\phi) \) using the identity \( \sin^2(\phi) + \cos^2(\phi) = 1 \). Thus, \( \cos^2(\phi) = 1 - \left(\frac{12}{13}\right)^2 = 1 - \frac{144}{169} = \frac{25}{169} \). Therefore, \( \cos(\phi) = \frac{5}{13} \).

Applying Cosine Addition Formula

To find \( \cos(\theta + \phi) \), use the cosine of angle sum formula: \( \cos(\theta + \phi) = \cos(\theta) \cos(\phi) - \sin(\theta) \sin(\phi) \). Substitute the values: \( \cos(\theta) = \frac{4}{5}, \cos(\phi) = \frac{5}{13}, \sin(\theta) = \frac{3}{5}, \sin(\phi) = \frac{12}{13} \).

Calculating \( \cos(\theta + \phi) \)

Substitute the values into the formula: \[\cos(\theta + \phi) = \left(\frac{4}{5}\right)\left(\frac{5}{13}\right) - \left(\frac{3}{5}\right)\left(\frac{12}{13}\right)\]. Simplifying gives \( \frac{20}{65} - \frac{36}{65} = \frac{-16}{65} \). Thus, \( \cos(\theta + \phi) = \frac{-16}{65} \).

Key concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are the opposites of the usual trigonometric functions. They help us find angles based on known side lengths of a right triangle. When given a ratio, such as the cosine or sine of an angle, these inverse functions give us the angle itself. For example, the notation \( \cos^{-1}\left(\frac{4}{5}\right) \) means that we are looking for an angle whose cosine is \( \frac{4}{5} \). The output is an angle, which we usually denote by a Greek letter like \( \theta \) or \( \phi \).
Similarly, \( \sin^{-1}\left(\frac{12}{13}\right) \) refers to the angle whose sine is \( \frac{12}{13} \). These angles are critical in various trigonometric calculations, such as those needed in this exercise to eventually find a specific cosine value through angle manipulation.

Angle Sum Formula

The angle sum formula is a key concept in trigonometry, used to find the trigonometric function of the sum of two angles. Specifically, the formula for the cosine of the sum of two angles \( \theta \) and \( \phi \) is given by:

\[ \cos(\theta + \phi) = \cos(\theta) \cos(\phi) - \sin(\theta) \sin(\phi) \]

This formula is extremely useful when you know the individual sines and cosines of the angles but need to determine the cosine of their sum. By applying this formula, you can easily find complex angle values based on simpler components. It's a handy technique widely used in trigonometry to simplify calculations and find solutions to problems like the one presented in the original exercise.

Trigonometric Functions

Trigonometric functions include sine, cosine, and tangent, among others. These functions relate the angles of a triangle to the lengths of its sides. The cosine function, for example, represents the ratio of the adjacent side to the hypotenuse in a right triangle. In the exercise, the cosine value \( \frac{4}{5} \) refers to an angle where the adjacent side is 4 units, and the hypotenuse is 5 units.
The sine function, on the other hand, represents the ratio of the opposite side to the hypotenuse. With \( \sin(\phi) = \frac{12}{13} \), this denotes an angle where the opposite side is 12, and the hypotenuse is 13. Understanding these fundamental relationships is crucial for solving many problems involving angles and triangles in trigonometry. Knowing these ratios can also help in deriving other important values, as demonstrated through the sine and cosine calculations in the exercise.

Algebraic Manipulation

Algebraic manipulation involves rearranging and simplifying expressions to find unknown values. It often includes using known identities and properties to simplify complex equations into manageable forms. In this exercise, we simplified trigonometric forms by calculating squares and using the identity \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
For instance, we calculated \( \sin(\theta) \) from \( \cos(\theta) \) using this identity, which is one essential tool in simplifying trigonometric expressions. Such algebraic techniques are indispensable in a wide range of situations, not just in trigonometry but in all areas of maths where complexities need reducing to simpler parts. This enables you to see through the solution clearly and accurately solve the problem at hand.