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Chapter 6: Problem 73

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

Short Answer

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

Step by step solution

01

Use the Trigonometric Identity

Recall the trigonometric identity for cosine of a sum: \[ \cos (A + B) = \cos A \cos B - \sin A \sin B \] Here, we can set \( A = \pi/2 \) and \( B = \alpha \).
02

Apply the Values

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

Simplify Using Known Values

We know that \( \cos(\pi/2) = 0 \) and \( \sin(\pi/2) = 1 \). Substituting these values into the equation gives: \[ \cos(\pi/2 + \alpha) = 0 \cdot \cos(\alpha) - 1 \cdot \sin(\alpha) = -\sin(\alpha) \]
04

Final Expression

Therefore, the expression simplifies to \[ \cos(\pi/2 + \alpha) = -\sin(\alpha) \]

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Key Concepts

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

Cosine Sum Identity
The cosine sum identity is a fundamental concept in trigonometry. It helps to simplify expressions that involve the sum of angles. This identity is particularly important because it breaks down more complex trigonometric expressions into simpler ones.

The identity is given by \(\cos (A + B) = \cos A \cos B - \sin A \sin B\). Here, A and B are any angles, and you can substitute any values to find the cosine of the sum of those angles.

Understanding this identity makes it easier to handle complex trigonometric equations. In our exercise, we used this identity to transform \(\cos (\pi / 2 + \alpha)\) into a simpler form.
Sine Function
The sine function is one of the basic trigonometric functions. It is defined as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle.

Mathematically, \(\sin( \theta ) = \frac{\text{opposite}}{\text{hypotenuse}}\).

Sine values range from -1 to 1 and are periodic with a period of \(2\pi\).

In our exercise, we used the value \(\sin (\pi/2) = 1\). This simplification was crucial for converting \(\cos (\pi / 2 + \alpha)\) into a simpler expression involving sine: \(\cos(\pi/2 + \alpha) = -\sin(\alpha)\).

Knowing these fundamental values and properties of the sine function makes solving trigonometric problems easier.
Trigonometric Simplification
Trigonometric simplification refers to the process of transforming a complex trigonometric expression into a simpler form. This often involves using identities like the cosine sum identity, sine function values, and known angles.

Simplification helps to make equations more manageable and easier to solve. For instance, in the given exercise, we started with \(\cos (\pi/2 + \alpha)\).

By applying the cosine sum identity and known trigonometric values, we simplified it to \(\cos(\pi/2 + \alpha) = -\sin(\alpha)\).

Key steps in trigonometric simplification often include:
  • Identifying relevant identities.
  • Substituting known values.
  • Reducing the expression step-by-step.
Mastering these techniques can make tackling trigonometric problems much less intimidating.

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Most popular questions from this chapter

Verify that each equation is an identity. $$\frac{1-\cos ^{2}\left(\frac{x}{2}\right)}{1-\sin ^{2}\left(\frac{x}{2}\right)}=\frac{1-\cos x}{1+\cos x}$$

Find all values of \(\alpha\) in \(\left[0^{\circ}, 360^{\circ}\right)\) that satisfy each equation. $$\tan \alpha=-\sqrt{3}$$

In each case, find \(\sin \alpha, \cos \alpha, \tan \alpha, \csc \alpha, \sec \alpha,\) and \(\cot \alpha\) $$\sin (2 \alpha)=-8 / 17 \text { and } 180^{\circ}<2 \alpha<270^{\circ}$$

Find the exact value of \(\tan (x / 2)\) given that \(\sin (x)=\sqrt{8 / 9}\) and \(3 \pi / 2

For each equation, either prove that it is an identity or prove that it is not an identity. $$\cot \frac{x}{2}-\tan \frac{x}{2}=\frac{\sin 2 x}{\sin ^{2} x}$$
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