Problem 126 - Precalculus

Question

For the following exercises, find the requested value. If \(\sin (t)=\frac{3}{8}\) and \(t\) is in the \(2^{\text { nd }}\) quadrant, find \(\cos (t)\)

Step-by-Step Solution

Answer

\( \cos(t) = -\frac{\sqrt{55}}{8} \) in the 2nd quadrant.

Step 1: Use the Pythagorean Identity

The Pythagorean identity states that for any angle \( t \), \( \sin^2(t) + \cos^2(t) = 1 \). To find \( \cos(t) \), we can substitute the value \( \sin(t) = \frac{3}{8} \) into this identity.

Step 2: Square the Sine Value

Calculate \( \sin^2(t) \) by squaring \( \frac{3}{8} \): \[ \sin^2(t) = \left(\frac{3}{8}\right)^2 = \frac{9}{64} \]

Step 3: Substitute in Pythagorean Identity

Substitute \( \sin^2(t) = \frac{9}{64} \) into the identity: \[ \sin^2(t) + \cos^2(t) = 1 \]\[ \frac{9}{64} + \cos^2(t) = 1 \]

Step 4: Solve for Cosine Squared

Rearrange to solve for \( \cos^2(t) \): \[ \cos^2(t) = 1 - \frac{9}{64} \]Convert 1 to \( \frac{64}{64} \) and subtract: \[ \cos^2(t) = \frac{64}{64} - \frac{9}{64} = \frac{55}{64} \]

Step 5: Determine the Sign of Cosine

Since \( t \) is in the second quadrant, the cosine of \( t \) is negative. Therefore, compute \( \cos(t) \):\[ \cos(t) = -\sqrt{\frac{55}{64}} \]

Step 6: Simplify the Cosine Value

Simplify \( \cos(t) \):\[ \cos(t) = -\frac{\sqrt{55}}{8} \]

Key Concepts

Pythagorean Identity

The Pythagorean Identity is a critical relationship in trigonometry. It expresses a fundamental connection between the sine and cosine of an angle. The identity is written as \( \sin^2(t) + \cos^2(t) = 1 \). This equation is akin to the Pythagorean Theorem and is valid for any angle \( t \). To understand this identity, remember it essentially says if you square the sine of an angle and add it to the square of the cosine of the same angle, the result is always one.
This identity is immensely useful because it allows one trigonometric function to be expressed in terms of another. For example, knowing \( \sin(t) \) can help you determine \( \cos(t) \), and vice versa, when using this identity. Applying it to the problem in our exercise, we started with \( \sin(t) = \frac{3}{8} \) and used the identity to find \( \cos(t) \). By squaring the sine value and substituting it in, we could solve for \( \cos^2(t) \) and subsequently \( \cos(t) \). This is a quintessential technique in solving trigonometric problems.

Trigonometric Functions

Trigonometric functions are a set of mathematical functions that relate the angles of a triangle to the lengths of its sides. The primary functions are sine, cosine, and tangent. They are crucial in describing oscillatory phenomena like waves.
- **Sine (\( \sin \)):** This function gives the ratio of the opposite side to the hypotenuse in a right triangle. In the coordinate plane, it represents the y-coordinate of a point on the unit circle.- **Cosine (\( \cos \)):** This function gives the ratio of the adjacent side to the hypotenuse. It corresponds to the x-coordinate on the unit circle.- **Tangent (\( \tan \)):** It is the ratio of sine to cosine. It represents the slope of a line from the origin to a point on the unit circle.
These functions are periodic, meaning they repeat their values over regular intervals. That's why they're frequently used in cycling or wave-like patterns. Given \( \sin(t) \) in the problem, we utilized this function to determine \( \cos(t) \), exploiting their interrelation via the Pythagorean Identity.

Second Quadrant

In trigonometry, the unit circle is divided into four quadrants. Each quadrant affects the sign of the trigonometric functions. In the second quadrant (angles between 90° and 180°, or \( \pi/2 \) and \( \pi \)), the values have specific sign characteristics:

• The sine function is positive.
• The cosine function is negative.
• The tangent function is negative.

Understanding these sign conventions is very important when solving trigonometric equations, as seen in our solution. When determining \( \cos(t) \) from \( \cos^2(t) = \frac{55}{64} \), we noted that \( t \) is in the second quadrant. Thus, \( \cos(t) \) should be negative, leading us to conclude that \( \cos(t) = -\frac{\sqrt{55}}{8} \).
Such quadrant rules prevent mistakes and guide correct evaluations of trigonometric values for given angles.