3 If tant: π = and 0 < t < , find sint, cost, sect, csct, cott. 4 2 Enter the exact answers. 4 sint= cost= sect = csct= cott = - - AYT Fal
3 If tant: π = and 0 < t < , find sint, cost, sect, csct, cott. 4 2 Enter the exact answers. 4 sint= cost= sect = csct= cott = - - AYT Fal
Trigonometry (11th Edition)
11th Edition
ISBN:9780134217437
Author:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Publisher:Margaret L. Lial, John Hornsby, David I. Schneider, Callie Daniels
Chapter1: Trigonometric Functions
Section: Chapter Questions
Problem 1RE:
1. Give the measures of the complement and the supplement of an angle measuring 35°.
Related questions
Question
![### Trigonometric Function Values from a Given Tangent
#### Problem:
If \( \tan t = \frac{3}{4} \) and \( 0 < t < \frac{\pi}{2} \), find the exact values of the following trigonometric functions:
- \( \sin t \)
- \( \cos t \)
- \( \sec t \)
- \( \csc t \)
- \( \cot t \)
#### Solution:
To solve this problem, use the given information and trigonometric identities to find the required values.
1. Start with the given tangent value:
\[ \tan t = \frac{3}{4} \]
2. For angles \( t \) in the first quadrant:
\[ \tan t = \frac{\text{opposite}}{\text{adjacent}} \]
Here, the opposite side is 3 and the adjacent side is 4 in a right triangle. Use the Pythagorean theorem to find the hypotenuse:
\[ \text{hypotenuse} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
3. Using the right triangle, find the other trigonometric functions:
\[ \sin t = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5} \]
\[ \cos t = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5} \]
\[ \sec t = \frac{1}{\cos t} = \frac{5}{4} \]
\[ \csc t = \frac{1}{\sin t} = \frac{5}{3} \]
\[ \cot t = \frac{1}{\tan t} = \frac{4}{3} \]
4. Enter the exact answers:
\[
\begin{aligned}
&\sin t = \frac{3}{5} \\
&\cos t = \frac{4}{5} \\
&\sec t = \frac{5}{4} \\
&\csc t = \frac{5}{3} \\
&\cot t = \frac{4}{3} \\
\end{aligned}
\]
*Note: The text boxes in the image are placeholders](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ffa18559e-82a9-4d38-b151-f799167474bb%2Fbb503ee6-f3da-4dda-b1fc-446950899dc0%2F2kaqk1_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Trigonometric Function Values from a Given Tangent
#### Problem:
If \( \tan t = \frac{3}{4} \) and \( 0 < t < \frac{\pi}{2} \), find the exact values of the following trigonometric functions:
- \( \sin t \)
- \( \cos t \)
- \( \sec t \)
- \( \csc t \)
- \( \cot t \)
#### Solution:
To solve this problem, use the given information and trigonometric identities to find the required values.
1. Start with the given tangent value:
\[ \tan t = \frac{3}{4} \]
2. For angles \( t \) in the first quadrant:
\[ \tan t = \frac{\text{opposite}}{\text{adjacent}} \]
Here, the opposite side is 3 and the adjacent side is 4 in a right triangle. Use the Pythagorean theorem to find the hypotenuse:
\[ \text{hypotenuse} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \]
3. Using the right triangle, find the other trigonometric functions:
\[ \sin t = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{3}{5} \]
\[ \cos t = \frac{\text{adjacent}}{\text{hypotenuse}} = \frac{4}{5} \]
\[ \sec t = \frac{1}{\cos t} = \frac{5}{4} \]
\[ \csc t = \frac{1}{\sin t} = \frac{5}{3} \]
\[ \cot t = \frac{1}{\tan t} = \frac{4}{3} \]
4. Enter the exact answers:
\[
\begin{aligned}
&\sin t = \frac{3}{5} \\
&\cos t = \frac{4}{5} \\
&\sec t = \frac{5}{4} \\
&\csc t = \frac{5}{3} \\
&\cot t = \frac{4}{3} \\
\end{aligned}
\]
*Note: The text boxes in the image are placeholders
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