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 sinθ=a\sin \theta = a and that θ\theta is an angle in the first quadrant. We are asked to evaluate cos(270θ)\cos(270^\circ - \theta) in terms of aa.

Step-by-step solution:

  1. Using the cosine identity: We can use the identity: cos(270θ)=sin(θ)\cos(270^\circ - \theta) = -\sin(\theta)

    This identity comes from the fact that subtracting an angle from 270270^\circ 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.

  2. Substitute sinθ=a\sin \theta = a: Since we are given that sinθ=a\sin \theta = a, we substitute this into the expression: cos(270θ)=a\cos(270^\circ - \theta) = -a

Thus, the value of cos(270θ)\cos(270^\circ - \theta) in terms of aa is a-a.


Would you like further clarification or a deeper explanation?

Here are five related questions to expand your understanding:

  1. How does the unit circle help in understanding angle transformations?
  2. What are the other trigonometric identities for angles involving 270270^\circ?
  3. How can you express sin(270θ)\sin(270^\circ - \theta) in terms of aa?
  4. What happens to the trigonometric ratios of angles in the different quadrants?
  5. Can we generalize the result for any angle of the form 360θ360^\circ - \theta?

Tip: Remember that angles subtracted from 270270^\circ or 9090^\circ 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