Math Problem Statement

Calculate the trigonometric expressions (cos(α), tan(α), sin(α+180°), cos(90°+α)) given that sin(α) = 0.8 and α > 90°.

Solution

The problem provided involves trigonometric functions and some calculations with angles. Here's the breakdown and solution:

Problem Analysis

We are given:

  • sin(α) = 0.8, with α > 90°.
  • We need to calculate the following trigonometric expressions: a. cos(α)
    b. tan(α)
    c. sin(α + 180°)
    d. cos(90° + α)

Solution

  1. Find cos(α):
    Using the Pythagorean identity: sin2(α)+cos2(α)=1\sin^2(α) + \cos^2(α) = 1 Substituting sin(α)=0.8\sin(α) = 0.8: (0.8)2+cos2(α)=1(0.8)^2 + \cos^2(α) = 1 0.64+cos2(α)=1    cos2(α)=0.360.64 + \cos^2(α) = 1 \quad \implies \quad \cos^2(α) = 0.36 cos(α)=±0.6\cos(α) = \pm 0.6 Since α>90α > 90^\circ, it is in the second quadrant, where cosine is negative. Thus: cos(α)=0.6\cos(α) = -0.6

  2. Find tan(α):
    The tangent function is given by: tan(α)=sin(α)cos(α)\tan(α) = \frac{\sin(α)}{\cos(α)} Substituting sin(α)=0.8\sin(α) = 0.8 and cos(α)=0.6\cos(α) = -0.6: tan(α)=0.80.6=43\tan(α) = \frac{0.8}{-0.6} = -\frac{4}{3}

  3. Find sin(α + 180°):
    Using the periodic property of sine: sin(α+180°)=sin(α)\sin(α + 180°) = -\sin(α) Substituting sin(α)=0.8\sin(α) = 0.8: sin(α+180°)=0.8\sin(α + 180°) = -0.8

  4. Find cos(90° + α):
    Using the co-function identity: cos(90°+α)=sin(α)\cos(90° + α) = -\sin(α) Substituting sin(α)=0.8\sin(α) = 0.8: cos(90°+α)=0.8\cos(90° + α) = -0.8

Final Answers:

a. cos(α)=0.6\cos(α) = -0.6
b. tan(α)=43\tan(α) = -\frac{4}{3}
c. sin(α+180°)=0.8\sin(α + 180°) = -0.8
d. cos(90°+α)=0.8\cos(90° + α) = -0.8


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Related Questions:

  1. How do trigonometric identities change in each quadrant?
  2. What is the periodicity of sine and cosine functions?
  3. How is the tangent function affected by angle shifts?
  4. Why does cosine become negative in the second quadrant?
  5. What are other co-function identities, such as sin(90°+α)\sin(90° + α)?

Tip:

Remember the ASTC rule (All Students Take Calculus) to determine the signs of trigonometric functions in different quadrants!

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Math Problem Analysis

Mathematical Concepts

Trigonometric functions
Pythagorean identity
Quadrants in trigonometry
Co-function identities

Formulas

sin²(α) + cos²(α) = 1
tan(α) = sin(α)/cos(α)
sin(α+180°) = -sin(α)
cos(90°+α) = -sin(α)

Theorems

Pythagorean identity
Co-function properties of sine and cosine
Quadrant-based sign rules (ASTC rule)

Suitable Grade Level

Grades 10-12