Math Problem Statement
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
-
Find cos(α):
Using the Pythagorean identity: Substituting : Since , it is in the second quadrant, where cosine is negative. Thus: -
Find tan(α):
The tangent function is given by: Substituting and : -
Find sin(α + 180°):
Using the periodic property of sine: Substituting : -
Find cos(90° + α):
Using the co-function identity: Substituting :
Final Answers:
a.
b.
c.
d.
Let me know if you'd like further details!
Related Questions:
- How do trigonometric identities change in each quadrant?
- What is the periodicity of sine and cosine functions?
- How is the tangent function affected by angle shifts?
- Why does cosine become negative in the second quadrant?
- What are other co-function identities, such as ?
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
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