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) =

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

### Step-by-step solution:

1. **Using the cosine identity**:
   We can use the identity:
   \[
   \cos(270^\circ - \theta) = -\sin(\theta)
   \]
   
   This identity comes from the fact that subtracting an angle from \(270^\circ\) 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 \(\sin \theta = a\)**:
   Since we are given that \(\sin \theta = a\), we substitute this into the expression:
   \[
   \cos(270^\circ - \theta) = -a
   \]

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

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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 \(270^\circ\)?
3. How can you express \(\sin(270^\circ - \theta)\) in terms of \(a\)?
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 \(360^\circ - \theta\)?

**Tip:** Remember that angles subtracted from \(270^\circ\) or \(90^\circ\) often involve using sine and cosine in complementary ways.

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) =

Explore how to evaluate cos(270° - θ) when sin θ = a for angles in the first quadrant. This problem uses key trigonometric identities and quadrant transformations. Suitable for high school students in Grades 10-12.

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