Problem 69 - Precalculus Mathematics for Calculus

Question

Find the values of the trigonometric functions of \(t\) from the given information. \(\tan t=-\frac{3}{4}, \quad \cos t > 0\)

Step-by-Step Solution

Answer

\( \sin t = -\frac{3}{5} \), \( \cos t = \frac{4}{5} \), \( \tan t = -\frac{3}{4} \).

Step 1: Determine the Quadrant

Since \( \tan t = -\frac{3}{4} \) and \( \cos t > 0 \), we need to identify the quadrant where \( \tan t \) is negative and \( \cos t \) is positive. The tangent is negative in the second and fourth quadrants. However, cosine is only positive in the fourth quadrant. Therefore, \( t \) is in the fourth quadrant.

Step 2: Express Trigonometric Functions in a Right Triangle

In the fourth quadrant, represent \( \tan t = -\frac{3}{4} \) using a right triangle. Here, for a reference angle \( \theta \), use opposite side = 3 and adjacent side = 4. The sign of the tangent, being negative, simply indicates that we are in the correct quadrant.

Step 3: Calculate the Hypotenuse

Use the Pythagorean theorem to find the hypotenuse \( r \). \( r = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \). Thus, the hypotenuse is 5.

Step 4: Find \( \sin t \) and \( \cos t \)

Now calculate the sine and cosine. Since \( \cos t \) is positive in the fourth quadrant, \( \cos t = \frac{4}{5} \). Since \( \sin t \) is negative in the fourth quadrant, \( \sin t = -\frac{3}{5} \).

Step 5: Confirm the Trigonometric Values

Double-check the trigonometric identities: \( \tan t = \frac{\sin t}{\cos t} = \frac{-\frac{3}{5}}{\frac{4}{5}} = -\frac{3}{4} \), which matches the given information. Ensure all values correspond to the chosen quadrant and satisfy the conditions.

Key Concepts

Quadrant

When solving trigonometric problems, the quadrant helps us determine the signs of trigonometric functions. The coordinate plane has four quadrants, each distinguished by different signs for sine, cosine, and tangent functions.

• The first quadrant has all positive signs.
• The second has sine positive, but cosine and tangent negative.
• The third has tangent positive, while sine and cosine are negative.
• The fourth quadrant, which is relevant here, has cosine positive and sine and tangent negative.

In the given problem, analyzing the signs of cosine and tangent helps pinpoint the right quadrant. Since cosine is positive and tangent is negative, we know the angle is in the fourth quadrant. This step is crucial to ensure we calculate the trigonometric functions with their correct signs.

Right Triangle

Using right triangles in trigonometry gives a visual way to understand the relationships between sides and angles. A right triangle consists of one angle measuring 90 degrees, creating relationships that help calculate functions.
In the given problem, the tangent function provides the ratio of the opposite side to the adjacent side of a right triangle. Represented as \( \tan t = -\frac{3}{4} \), it suggests imagining a right triangle where:

• The opposite side measures 3.
• The adjacent side measures 4.

The negative tangent signifies the particular quadrant (fourth) rather than modifying triangle dimensions. Drawing the triangle helps visualize these ratios and assists in determining the hypotenuse using the Pythagorean theorem.

Pythagorean Theorem

The Pythagorean theorem is a fundamental concept connecting the three sides of a right triangle with a 90-degree angle. The theorem states that the square of the hypotenuse equals the sum of the squares of the other two sides. Mathematically, it's written as \( a^2 + b^2 = c^2 \), where \(c\) is the hypotenuse.
In our problem, with side lengths of 3 and 4, the theorem helps find the hypotenuse:

• \(3^2 + 4^2 = 9 + 16 = 25\)
• The hypotenuse is \(\sqrt{25} = 5\).

This calculation confirms that the constructed right triangle is accurate. Utilizing the Pythagorean theorem ensures we have the correct hypotenuse necessary for further trigonometric calculations.

Sine and Cosine

Sine and cosine are two main functions in trigonometry, describing the ratios of specific sides of a right triangle in relation to an angle. Most notably:

• Sine (\( \sin \)) is the ratio of the opposite side to the hypotenuse.
• Cosine (\( \cos \)) is the ratio of the adjacent side to the hypotenuse.

In the fourth quadrant, where our angle is located, these functions have:

• Cosine positive, meaning \( \cos t = \frac{4}{5} \)
• Sine negative, giving \( \sin t = -\frac{3}{5} \)

These calculations align with our right triangle sides: opposite as 3, adjacent as 4, and hypotenuse as 5. It's important to keep in mind the quadrant, as the signs of these functions can vary significantly.