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Chapter 3: Problem 88

Write each expression as a function of \(\alpha\) alone. $$ \cos (\alpha-\pi / 2) $$

Short Answer

Expert verified
\( \cos ( \alpha - \pi / 2 ) = \sin \alpha \)

Step by step solution

01

Recall the trigonometric identity

Recall the trigonometric identity for the cosine of a difference: \[ \cos(A - B) = \cos A \cos B + \sin A \sin B \]
02

Substitute the given values

Substitute \(A = \alpha\) and \(B = \pi / 2\) in the identity from Step 1: \[ \cos ( \alpha - \pi / 2 ) = \cos \alpha \cos( \pi / 2 ) + \sin \alpha \sin( \pi / 2 ) \]
03

Simplify using known values

Use known values of \( \cos (\pi / 2) = 0 \) and \( \sin (\pi / 2) = 1 \): \[ \cos ( \alpha - \pi / 2 ) = \cos \alpha (0) + \sin \alpha (1) \]
04

Final simplification

Simplify the expression: \[ \cos ( \alpha - \pi / 2 ) = 0 + \sin(\alpha) \]\[ \cos ( \alpha - \pi / 2 ) = \sin \alpha \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Cosine Difference Formula
Understanding the cosine difference formula is important for solving various trigonometric problems. The formula states: \[ \cos(A - B) = \cos A \cos B + \sin A \sin B \] This formula helps break down the cosine of a difference into simpler sine and cosine terms.
To apply this effectively:
  • Identify the angles A and B.
  • Substitute the angles into the formula.
  • Simplify using known values of sine and cosine for standard angles.
For the given exercise, we set A equal to \(\alpha\) and B to \(\pi / 2\). By substituting into the formula, we can transform \(\cos(\alpha - \pi / 2)\) into expressions involving \(\cos \alpha\) and \(\sin \alpha\).
This formula is especially useful because it translates a more complex trigonometric expression into simpler components.
Sine and Cosine Values
When using trigonometric identities, it's important to remember key sine and cosine values for standard angles like 0, \(\pi / 2\), and \(\pi\) radians. Specific values you should know:
  • \(\cos(0) = 1\)
  • \(\cos(\pi / 2) = 0\)
  • \(\cos(\pi) = -1\)
  • \(\sin(0) = 0\)
  • \(\sin(\pi / 2) = 1\)
  • \(\sin(\pi) = 0\)

In the exercise:
  • We use \(\cos(\pi / 2) = 0\) and \(\sin(\pi / 2) = 1\) to simplify our expression.
  • By replacing these values in our equation, we make our calculations manageable.
Knowing these values by heart allows for quick and accurate simplifications.
Angle Conversion
Understanding angle conversion is crucial in trigonometry. Angles can be measured in degrees or radians. To convert between them, remember: \[ \text{Degrees} = \frac{180}{\pi} \times \text{Radians} \ta\quad\quad \text{Radians} = \frac{\pi}{180} \times \text{Degrees} \]
For example:
  • 90 degrees converts to \(\pi / 2\) radians.
  • 180 degrees converts to \(\pi\) radians.
In our exercise, recognizing that \(\pi / 2\) radians is equivalent to 90 degrees helps in visualizing why certain sine and cosine values are used. The ability to switch between degrees and radians aids in solving and understanding trigonometric problems better.
Keep practicing these conversions to become more comfortable with both measurement systems.

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Most popular questions from this chapter

Solve each problem. Find the exact value of \(\cos (2 \alpha)\) given that \(\sin (\alpha)=8 / 17\) and \(\alpha\) is in quadrant II.

Prove that each equation is an identity. \(2 \sin ^{2}\left(\frac{u}{2}\right)=\frac{\sin ^{2} u}{1+\cos u}\)

State the three Pythagorean identities.

Explain why \(\tan (2 \alpha)=2 \tan (\alpha)\) is not an identity by using graphs and by using the definition of the tangent function.

Write each expression as a function of \(\alpha\) alone. $$ \cos \left(\alpha-360^{\circ}\right) $$
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