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Chapter 20: Problem 18

Simplify the given expressions. $$\sin (x+\pi / 2)$$

Short Answer

Expert verified
The expression \( \sin(x + \pi/2) \) simplifies to \( \cos x \).

Step by step solution

01

Use of Trigonometric Identity

To simplify the expression \( \sin(x + \pi/2) \), we use the trigonometric identity: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). Let \( a = x \) and \( b = \pi/2 \).
02

Calculating Sine and Cosine Values

We know \( \cos(\pi/2) = 0 \) and \( \sin(\pi/2) = 1 \). Substitute these values into the identity: \( \sin(x + \pi/2) = \sin x \cdot 0 + \cos x \cdot 1 \).
03

Simplify the Expression

Simplify the expression by multiplying: \( \sin(x + \pi/2) = 0 + \cos x = \cos x \). Therefore, the simplified expression is \( \cos x \).

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Key Concepts

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

Simplifying Trigonometric Expressions
Simplifying trigonometric expressions can sometimes seem tricky, but it's all about applying the right trigonometric identities and reducing the expression to its simplest form. The goal is to make the expression easier to work with and understand.

Here's a quick guide on how to approach such expressions:
  • Identify the trigonometric identity that might apply to your expression. This could involve sum and difference formulas, double angle formulas, or basic identities.
  • Substitute and simplify, ensuring you correctly apply any known trigonometric values like \( \sin(\pi/2) = 1 \) or \( \cos(\pi/2) = 0 \).
  • Reduce the expression to simplest terms by performing basic arithmetic operations.
Following these steps, as we did with \( \sin(x + \pi/2) \), can make even complex expressions much simpler.
Sine and Cosine Functions
The sine and cosine functions are foundational concepts in trigonometry, often related to angles of a right triangle. These functions are periodic and oscillate between -1 and 1.

Key features include:
  • Sine Function (\(\sin\)): It represents the y-coordinate (height) on the unit circle for a given angle. The function starts at 0, rises to 1 at \(\pi/2\), and falls back down.
  • Cosine Function (\(\cos\)): It represents the x-coordinate (horizontal position) on the unit circle for a given angle. It starts at 1, decreases to 0 at \(\pi/2\), and then goes to -1 at \(\pi\).
  • Both functions are used in various trigonometric identities and can be expressed in terms of each other using formulas like \( \sin(x + \pi/2) = \cos x \).
Understanding these functions helps in visualizing how angles relate to the coordinates on a circle and also in solving trigonometric equations.
Angle Addition Formulas
Angle addition formulas are valuable tools in trigonometry used to determine the sine, cosine, or tangent of the sum or difference of two angles. These formulas provide a way to break down complex expressions into simpler terms.

For sine and cosine, the formulas are:
  • \( \sin(a + b) = \sin a \cos b + \cos a \sin b \)
  • \( \cos(a + b) = \cos a \cos b - \sin a \sin b \)
By using these formulas, you can simplify trigonometric expressions like \( \sin(x + \pi/2) \) by substituting known values for \( \sin(\pi/2) \) and \( \cos(\pi/2) \). These identities are essential for solving many problems in trigonometry that involve adding or subtracting angles, and they highlight the interrelation between different trigonometric functions.

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