Problem 43 - Contemporary Precalculus

Question

Find the exact functional value without using a calculator. $$\cos \left[\sin ^{-1}(\sqrt{3} / 5)\right]$$

Step-by-Step Solution

Answer

Answer: The exact value is $\frac{\sqrt{22}}{5}$.

Step 1: Write down the problem

The given expression is: $$\cos \left[\sin ^{-1}(\sqrt{3} / 5)\right]$$ We need to find the exact value without using a calculator.

Step 2: Identify the angle

We know that \(\sin ^{-1} (\sqrt{3} / 5)\) is the angle whose sine value is \(\sqrt{3} / 5\). Let's call this angle \(\theta\). So, we have: $$\sin(\theta) = \frac{\sqrt{3}}{5}$$

Step 3: Use the Pythagorean theorem

Recall the Pythagorean theorem for a right triangle with sides a, b, and c, and angle \(\theta\) opposite side a: $$a^2 + b^2 = c^2$$ In our case, a is the side opposite to angle \(\theta\), so we have: $$\left(\frac{\sqrt{3}}{5}\right)^2 + b^2 = 1^2$$

Step 4: Solve for the missing side

Now, solve for b: $$\frac{3}{25} + b^2 = 1$$ $$b^2 = 1 - \frac{3}{25} = \frac{22}{25}$$ $$b = \sqrt{\frac{22}{25}} = \frac{\sqrt{22}}{5}$$

Step 5: Find the cosine of the angle

Since we have found the adjacent side (b), we can use the definition of cosine for a right triangle to find the cosine of the angle \(\theta\): $$\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{b}{1} = \frac{\sqrt{22}}{5}$$

Step 6: Write the final answer

Replacing the cosine function with our found value, we get the final answer: $$\cos \left[\sin ^{-1}(\sqrt{3} / 5)\right] = \frac{\sqrt{22}}{5}$$

Key Concepts

Inverse Trigonometric Functions

Inverse trigonometric functions are essential when we want to find the angle of a right triangle given one of its sides' ratios. For instance, \(\sin^{-1}(x)\) or arcsin is used to find the angle whose sine value is \(x\). In our problem, \(\sin^{-1}(\sqrt{3} / 5)\) gives us an angle \(\theta\) where its sine is exactly \(\sqrt{3} / 5\). This process involves reversing the action of the sine function to determine the specific angle rather than calculating the sine of a given angle.

• The angle \(\theta\) from \(\sin^{-1}(\sqrt{3} / 5)\) is the one that satisfies the equation \(\sin(\theta) = \sqrt{3} / 5\).
• This technique is widely used in fields such as engineering and physics to solve problems involving angles and distances.

Understanding these functions enhances our ability to solve trigonometric problems by working backward from known sine, cosine, or tangent values.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in mathematics, mainly concerned with right triangles. It states that in a right triangle, the square of the length of the hypotenuse (side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. Mathematically, this is expressed as \(a^2 + b^2 = c^2\).

In our context, we used this theorem to find the missing side of a hypothetical right triangle where \(a\) is \(\sqrt{3} / 5\), \(b\) is the unknown side, and the hypotenuse is 1 (a unit circle scenario). By applying the theorem:

• We square the given side: \(\left(\frac{\sqrt{3}}{5}\right)^2 = \frac{3}{25}\).
• Then arrange the equation to solve for the missing side: \(b^2 = 1 - \frac{3}{25} = \frac{22}{25}\).
• Finally, solve for \(b\) by taking the square root: \(b = \frac{\sqrt{22}}{5}\).

Being familiar with the Pythagorean theorem allows us to uncover unknown sides and discover essential features of right triangles.

Right Triangle Trigonometry

Right triangle trigonometry is a powerful tool that helps us determine the relationships between the angles and sides of a right triangle. This involves using the sine, cosine, and tangent functions, which relate the angles to side lengths.

In the right triangle given in the exercise:

• The sine function of angle \(\theta\) is the ratio of the opposite side to the hypotenuse: \(\sin(\theta) = \frac{\sqrt{3}}{5}\).
• With the use of the adjacent side found through the Pythagorean theorem, the cosine can be calculated as: \(\cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{\sqrt{22}}{5}\).

Understanding right triangle trigonometry allows us to apply these principles to find unknown angles or side lengths in various geometrical problems. Furthermore, it forms the foundation for more complex trigonometric applications found in higher mathematics.