Math Problem Statement

User responses cleared Homework:8.5 Question 9, 8.5.37 Part 1 of 4 HW Score: 61.54%, 8 of 13 points Points: 0 of 1

Skip to Main content Question 1 Question 2 Question 3 Question 4 Question 5 Question 6 Question 7 Question 8 Question 9 Question 10 Question 11 Question 12 Question 13 Question content area top Part 1 Find the exact value of each of the following under the given conditions below. tangent alpha equals negative StartFraction 12 Over 5 EndFraction comma StartFraction pi Over 2 EndFraction less than alpha less than pi ​; sine beta equals StartFraction StartRoot 3 EndRoot Over 2 EndFraction comma 0 less than beta less than StartFraction pi Over 2 EndFraction ​(a) sine left parenthesis alpha plus beta right parenthesis        ​(b) cosine left parenthesis alpha plus beta right parenthesis        ​(c) sine left parenthesis alpha minus beta right parenthesis        ​(d) tangent left parenthesis alpha minus beta right parenthesis

Solution

To solve this problem, we will use the given values of tan⁡(α)=−125\tan(\alpha) = -\frac{12}{5} and sin⁡(β)=32\sin(\beta) = \frac{\sqrt{3}}{2} within the specified ranges for α\alpha and β\beta, and we will apply the sum and difference formulas for sine, cosine, and tangent.

Given Information:

  • tan⁡(α)=−125\tan(\alpha) = -\frac{12}{5}, with π2<α<π\frac{\pi}{2} < \alpha < \pi (so α\alpha is in the second quadrant).
  • sin⁡(β)=32\sin(\beta) = \frac{\sqrt{3}}{2}, with 0<β<π20 < \beta < \frac{\pi}{2} (so β\beta is in the first quadrant).

Step 1: Find sin⁡(α)\sin(\alpha) and cos⁡(α)\cos(\alpha)

We can use the identity (\tan(\alpha)

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Sum and Difference Formulas
Sine
Cosine
Tangent

Formulas

sin(α + β) = sin(α)cos(β) + cos(α)sin(β)
cos(α + β) = cos(α)cos(β) - sin(α)sin(β)
sin(α - β) = sin(α)cos(β) - cos(α)sin(β)
tan(α - β) = (tan(α) - tan(β)) / (1 + tan(α)tan(β))

Theorems

Sum and Difference Identities for Trigonometric Functions

Suitable Grade Level

Grades 10-12

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