Problem 52 - Algebra and Trigonometry

Question

Use a sketch to find the exact value of each expression. $$ \cos \left[\sin ^{-1}\left(-\frac{4}{5}\right)\right] $$

Step-by-Step Solution

Answer

The exact value of \(\cos [\sin^{-1}(-4/5)]\) is \(\frac{3}{5}\).

Step 1: Understanding the task

The inverse sine function, \(\sin^{-1}(x)\), outputs an angle whose sine is \(x\). So, \(\sin^{-1}(-\frac{4}{5})\) outputs an angle whose sine is \(-\frac{4}{5}\). This lies in the third or fourth quadrant where the sine values are negative.

Step 2: Sketch and identify the triangle

Sketch a right triangle in the fourth quadrant, where the opposite side is -4 (since sine is negative), and the hypotenuse is 5. The adjacent side can be found using the Pythagorean theorem, \(a^2 + b^2 = c^2\), solving for \(a\), you get \(\sqrt{5^2 - (-4)^2} = 3\).

Step 3: Find Cosine value

Cosine of this angle can be represented as the ratio of the adjacent side to the hypotenuse. In this case, \(\cos(\sin^{-1}(-\frac{4}{5}))\) is the ratio of 3 to 5, which is \(\frac{3}{5}\).

Key Concepts

Inverse Sine Function

The inverse sine function, often written as \(\sin^{-1}(x)\) or \(\arcsin(x)\), plays a crucial role in determining angles associated with a particular sine value in trigonometry. When we come across an expression like \(\sin^{-1}(-\frac{4}{5})\), we are essentially looking for an angle whose sine is \(\frac{-4}{5}\). It's important to remember that the sine function represents the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle.

The domain of the inverse sine function is restricted to \( -1 \leq x \leq 1 \) because sine values can only range between -1 and 1. Correspondingly, the range of the inverse sine function (the angles it can output) is limited to \( -\frac{\pi}{2} \leq \theta \leq \frac{\pi}{2} \), which corresponds to the angles in the first and fourth quadrants for positive values of \(x\), and the angles in the first and second quadrants for negative values of \(x\). Since our value of \(\sin^{-1}(-\frac{4}{5})\) is negative, the angle lies in either the third or the fourth quadrant where sine values are negative.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry that relates the sides of a right-angled triangle. It states that in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. This can be expressed in the formula \(a^2 + b^2 = c^2\), where \(a\) and \(b\) are the lengths of the legs, and \(c\) is the length of the hypotenuse.

In the context of the provided exercise, we utilized this theorem to find the length of the adjacent side (often referred to as the 'base' in the context of a triangle), given the opposite side is -4 and the hypotenuse is 5. By re-arranging the Pythagorean theorem, we find that the adjacent side \(a\) has a length which can be calculated using \(a = \sqrt{c^2 - b^2}\). Substituting our given values yields \(a = \sqrt{5^2 - (-4)^2} = \sqrt{25 - 16} = \sqrt{9} = 3\). This step was critical in enabling us to find the required trigonometric ratio for the cosine function.

Trigonometric Ratios

Trigonometric ratios are relationships between the sides of a right-angled triangle that are used to define the trigonometric functions. These ratios, including sine, cosine, and tangent, depend on the particular angle in question. The cosine of an angle, symbolized as \(\cos(\theta)\), is the ratio of the adjacent side to the hypotenuse in a right-angled triangle.

In the exercise, after determining the adjacent side's length using the Pythagorean theorem, we need to apply our knowledge of trigonometric ratios to find the cosine of the angle given by \(\sin^{-1}(-\frac{4}{5})\). Here, \(\cos(\sin^{-1}(-\frac{4}{5})) = \frac{adjacent}{hypotenuse} = \frac{3}{5}\), which is the exact value being sought. This demonstrates not only the practical application of trigonometric ratios but also how various elements of trigonometry, such as inverse functions and the Pythagorean theorem, are intertwined in solving problems.