Use the figure shown to the right and the given values to find​ c, sine theta​, and tangent theta. 
aequals7​, bequals12
theta
alpha
b
c
a
Question content area bottom
Part 1
cequals
  
enter your response here   ​(Type an integer or decimal rounded to three decimal places as​ needed.)
Part 2
sine thetaequals
  
enter your response here ​(Type an integer or decimal rounded to three decimal places as​ needed.)
Part 3
tangent thetaequals
  
enter your response here  ​(Type an integer or decimal rounded to three decimal places as​ needed.)

To solve this problem, we'll use the Pythagorean theorem, sine, and tangent trigonometric functions based on the provided information. Since no diagram was provided, I'll assume this refers to a right triangle where \( a = 7 \) and \( b = 12 \) are the lengths of the two legs, with \( c \) being the hypotenuse. Theta (θ) is the angle opposite side \( a \), while alpha (α) is opposite side \( b \).

### Part 1: Finding \( c \) (the hypotenuse)
According to the Pythagorean theorem:
\[
c = \sqrt{a^2 + b^2}
\]
Substitute the given values of \( a \) and \( b \):
\[
c = \sqrt{7^2 + 12^2} = \sqrt{49 + 144} = \sqrt{193} \approx 13.892
\]

Thus, \( c \approx 13.892 \).

### Part 2: Finding \( \sin(\theta) \)
The sine of an angle in a right triangle is the ratio of the opposite side to the hypotenuse:
\[
\sin(\theta) = \frac{a}{c}
\]
Substitute the known values of \( a \) and \( c \):
\[
\sin(\theta) = \frac{7}{13.892} \approx 0.504
\]

Thus, \( \sin(\theta) \approx 0.504 \).

### Part 3: Finding \( \tan(\theta) \)
The tangent of an angle is the ratio of the opposite side to the adjacent side:
\[
\tan(\theta) = \frac{a}{b}
\]
Substitute the known values of \( a \) and \( b \):
\[
\tan(\theta) = \frac{7}{12} \approx 0.583
\]

Thus, \( \tan(\theta) \approx 0.583 \).

### Final Answers:
1. \( c \approx 13.892 \)
2. \( \sin(\theta) \approx 0.504 \)
3. \( \tan(\theta) \approx 0.583 \)

Would you like more details or further clarifications on any of the steps?

Here are 5 related questions:
1. How do you calculate \( \cos(\theta) \) in a right triangle?
2. What is the relationship between sine, cosine, and tangent in a right triangle?
3. How can you verify the Pythagorean theorem using given side lengths?
4. What are the other trigonometric ratios (like \( \cot(\theta) \), \( \sec(\theta) \), \( \csc(\theta) \)) for this triangle?
5. How do you calculate the angle \( \theta \) from the tangent value?

**Tip:** Memorizing the basic trigonometric ratios for common angles like 30°, 45°, and 60° can help you solve problems more quickly.

Use the figure shown to the right and the given values to find c, sine theta, and tangent theta. a = 7, b = 12.

Learn how to calculate the hypotenuse, sine of theta, and tangent of theta for a right triangle where a = 7 and b = 12. This problem applies the Pythagorean theorem and trigonometric functions, ideal for students in Grades 9-11.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation