Math Problem Statement
Let θ be an angle in the first quadrant with sin θ = a . See figure to the right. Evaluate the following terms of a. cos(270° - θ) =
Solution
We are given that and that is an angle in the first quadrant. We are asked to evaluate in terms of .
Step-by-step solution:
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Using the cosine identity: We can use the identity:
This identity comes from the fact that subtracting an angle from shifts the reference angle into the third quadrant, where cosine is negative, and the sine of the angle is equivalent to the cosine of its complement.
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Substitute : Since we are given that , we substitute this into the expression:
Thus, the value of in terms of is .
Would you like further clarification or a deeper explanation?
Here are five related questions to expand your understanding:
- How does the unit circle help in understanding angle transformations?
- What are the other trigonometric identities for angles involving ?
- How can you express in terms of ?
- What happens to the trigonometric ratios of angles in the different quadrants?
- Can we generalize the result for any angle of the form ?
Tip: Remember that angles subtracted from or often involve using sine and cosine in complementary ways.
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Math Problem Analysis
Mathematical Concepts
Trigonometry
Angle Transformation
Unit Circle
Formulas
cos(270° - θ) = -sin(θ)
Theorems
Trigonometric identities
Quadrant angle transformations
Suitable Grade Level
Grades 10-12
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