Find all the missing side lengths and angle measures of each triangle. 1. 16 60 2.
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°.
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Question
![### Triangle Problem Solving
#### Instructions
Find all the missing side lengths and angle measures of each triangle.
#### Problem 1
Given Triangle: \(\triangle ACT\)
**Properties:**
- \( \angle A \) is a right angle (\(90^\circ\)).
- \( \angle C = 60^\circ \).
- Side \( TC = 16 \).
**Diagram Description:**
- The triangle \( \triangle ACT \) has point \( A \) as the right angle.
- Side \( AC \) is the base.
- Side \( AT \) is the height (unknown).
- Side \( TC = 16 \) (hypotenuse).
- Angle \( \angle C \) measures \( 60^\circ \).
#### Steps to Solve
1. **Find \( \angle T \)**
- Since \( \angle A \) is \( 90^\circ \) and \( \angle C \) is \( 60^\circ \),
- \( \angle T = 90^\circ - 60^\circ = 30^\circ \).
2. **Use Trigonometry to Find Missing Sides**
- Use the sine and cosine ratios appropriate for the given angles.
3. **Calculate Side \( AT \) (opposite to \( \angle C \))**
- \( \sin 60^\circ = \frac{\text{opposite}}{\text{hypotenuse}} \)
- \( \sin 60^\circ = \frac{AT}{16} \)
- \( AT = 16 \times \sin 60^\circ = 16 \times \frac{\sqrt{3}}{2} \approx 13.86 \)
4. **Calculate Side \( AC \) (adjacent to \( \angle C \))**
- \( \cos 60^\circ = \frac{\text{adjacent}}{\text{hypotenuse}} \)
- \( \cos 60^\circ = \frac{AC}{16} \)
- \( AC = 16 \times \cos 60^\circ = 16 \times \frac{1}{2} = 8 \)
**Final Values:**
- \( AT \approx 13.86 \)
- \( AC = 8 \)
- \( \angle T = 30^\circ \)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F8592bc7f-c6b6-4f68-857a-4219f7ee1e70%2F589f3d14-a327-4eaf-b0ff-ae2ace61dfdc%2F3396z86_processed.jpeg&w=3840&q=75)
Transcribed Image Text:### Triangle Problem Solving
#### Instructions
Find all the missing side lengths and angle measures of each triangle.
#### Problem 1
Given Triangle: \(\triangle ACT\)
**Properties:**
- \( \angle A \) is a right angle (\(90^\circ\)).
- \( \angle C = 60^\circ \).
- Side \( TC = 16 \).
**Diagram Description:**
- The triangle \( \triangle ACT \) has point \( A \) as the right angle.
- Side \( AC \) is the base.
- Side \( AT \) is the height (unknown).
- Side \( TC = 16 \) (hypotenuse).
- Angle \( \angle C \) measures \( 60^\circ \).
#### Steps to Solve
1. **Find \( \angle T \)**
- Since \( \angle A \) is \( 90^\circ \) and \( \angle C \) is \( 60^\circ \),
- \( \angle T = 90^\circ - 60^\circ = 30^\circ \).
2. **Use Trigonometry to Find Missing Sides**
- Use the sine and cosine ratios appropriate for the given angles.
3. **Calculate Side \( AT \) (opposite to \( \angle C \))**
- \( \sin 60^\circ = \frac{\text{opposite}}{\text{hypotenuse}} \)
- \( \sin 60^\circ = \frac{AT}{16} \)
- \( AT = 16 \times \sin 60^\circ = 16 \times \frac{\sqrt{3}}{2} \approx 13.86 \)
4. **Calculate Side \( AC \) (adjacent to \( \angle C \))**
- \( \cos 60^\circ = \frac{\text{adjacent}}{\text{hypotenuse}} \)
- \( \cos 60^\circ = \frac{AC}{16} \)
- \( AC = 16 \times \cos 60^\circ = 16 \times \frac{1}{2} = 8 \)
**Final Values:**
- \( AT \approx 13.86 \)
- \( AC = 8 \)
- \( \angle T = 30^\circ \)
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