Problem 30 - Precalculus Essentials

Question

\(0 \leq t<\frac{\pi}{2}\) and \(\sin t\) is given. Use the Pythagorean identity \(\sin ^{2} t+\cos ^{2} t=1\) to find \(\cos t\) $$\sin t=\frac{7}{8}$$

Step-by-Step Solution

Answer

The value of \( \cos t \) when \( \sin t=\frac{7}{8} \) and \( 0 \leq t<\frac{\pi}{2} \) is \( \cos t = \sqrt{1 - \left(\frac{7}{8}\right)^2} \).

Step 1: Identify the given value

From the given exercise, we know the value of \( \sin t \) which is \( \frac{7}{8} \).

Step 2: Use the Pythagorean identity

Using the Pythagorean identity \( \sin ^{2} t+\cos ^{2} t=1 \), we can replace \( \sin t \) with the given value to solve for \( \cos t \). So, the equation would be \( \left(\frac{7}{8}\right)^{2}+\cos^{2} t =1.

Step 3: Solve for \( \cos t \)

Subtract \(\left(\frac{7}{8}\right)^{2}\) from both sides of the equation to isolate \( \cos^2 t \) on one side. This gives us: \( \cos^2 t = 1 - \left(\frac{7}{8}\right)^2 \).

Step 4: Compute \( \cos t \)

To solve for \( \cos t \), take the square root of both sides: \( \cos t = \sqrt{1 - \left(\frac{7}{8}\right)^2} \).

Step 5: Determine the sign of \( \cos t \)

Since \( t \) is given to be in the range \( 0 \leq t<\frac{\pi}{2} \), which falls in the first quadrant where cosine values are positive, we consider the positive square root, \( \cos t = \sqrt{1 - \left(\frac{7}{8}\right)^2} \).

Key Concepts

Trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the angles and sides of triangles, especially right triangles. In this realm, understanding the basic trigonometric functions such as sine, cosine, and tangent is essential. Each of these functions represents a ratio of different sides of a right triangle.
For example:

• **Sine (\( \sin \)):** The ratio of the length of the opposite side to the hypotenuse.
• **Cosine (\( \cos \)):** The ratio of the length of the adjacent side to the hypotenuse.
• **Tangent (\( \tan \)):** The ratio of the length of the opposite side to the adjacent side.

These functions are fundamental in trigonometry and are used in various applications, from basic geometry to complex calculus. When you understand these concepts, you can solve many trigonometric problems, including those involving angles and distances.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are always true. These identities are crucial in simplifying and transforming expressions in trigonometry. One of the most famous and useful identities is the Pythagorean identity, which is given by:\[sin^2 t + \cos^2 t = 1\]This identity allows you to express one trigonometric function in terms of another. For example, knowing \( \sin t \), you can find \( \cos t \) using this identity by rearranging it to \( \cos^2 t = 1 - \sin^2 t \).
This concept serves as a tool for verifying and deriving other identities. By mastering these core identities, students can solve a wide array of trigonometric equations more efficiently.

Solving Trigonometric Equations

To solve trigonometric equations, like finding the cosine value given the sine value, you use identities and properties of functions. In our exercise example, we know \( \sin t = \frac{7}{8} \) and we need to find \( \cos t \) when \( 0 \leq t < \frac{\pi}{2} \).
The process involves:

• Substitute the known sine value into the Pythagorean identity: \( (\frac{7}{8})^2 + \cos^2 t = 1 \).
• Simplify: \( \cos^2 t = 1 - (\frac{7}{8})^2 \).
• Calculate: \( \cos t = \sqrt{1 - (\frac{7}{8})^2} \).

Since \( t \) is in the first quadrant (where all trigonometric functions are positive), \( \cos t \) is positive. This approach to solving ensures accuracy by applying mathematical properties and aids in tackling more complex problems involving trigonometric equations.