/*! This file is auto-generated */ .wp-block-button__link{color:#fff;background-color:#32373c;border-radius:9999px;box-shadow:none;text-decoration:none;padding:calc(.667em + 2px) calc(1.333em + 2px);font-size:1.125em}.wp-block-file__button{background:#32373c;color:#fff;text-decoration:none} Problem 57 Establish each identity. $$ ... [FREE SOLUTION] | 91影视

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Chapter 8: Problem 57

Establish each identity. $$ \sin \left(\frac{3 \pi}{2}+\theta\right)=-\cos \theta $$

Short Answer

Expert verified
Using the sine addition formula and known values, we get \( \text{sin}\big(\frac{3\text{蟺}}{2}+胃\big) = -\text{cos}(胃) \).

Step by step solution

01

Identify the Trigonometric Formula

The given problem involves an angle addition identity. Specifically, it uses the sine addition formula: \[ \frac{3\frac{蟺}{2}} + 胃 \]
02

Use the Sine Addition Formula

The sine addition formula is: \( \text{sin}(a + b) = \text{sin}(a) \text{cos}(b) + \text{cos}(a) \text{sin}(b) \)Here, \( a = \frac{3 \text{蟺}}{2} \) and \( b = 胃 \).
03

Substitute and Simplify

Using the known values:\( \text{sin}\big(\frac{3\text{蟺}}{2}\big) = -1 \) and \( \text{cos}\big(\frac{3\text{蟺}}{2}\big) = 0 \),the identity becomes: \[ \text{sin}\big(\frac{3\text{蟺}}{2}+胃\big) = (-1)\text{cos}(胃) + 0\text{sin}(胃) = -\text{cos}(胃) \]
04

Conclude the Identity

As a result of the simplification, we've established that: \[ \text{sin}\big(\frac{3\text{蟺}}{2}+胃\big) = -\text{cos}(胃) \]

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Key Concepts

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

sine addition formula
The sine addition formula is a fundamental tool in trigonometry. It allows us to express the sine of a sum of two angles in terms of the sines and cosines of the individual angles. The formula is:
\(\sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b)\)
Using this formula, we can break down complex trigonometric expressions into simpler ones.
Consider the given exercise: \(\sin \left( \frac{3 \pi}{2} + \theta \right)\). Here, our angles are \(a = \frac{3 \pi}{2}\) and \(b = \theta\). Applying the sine addition formula, we get:
  • \(\sin \left( \frac{3 \pi}{2} + \theta \right) = \sin \left( \frac{3 \pi}{2} \right) \cos(\theta) + \cos \left( \frac{3 \pi}{2} \right) \sin(\theta)\)

This expression uses the known sine and cosine values for the angle \(\frac{3 \pi}{2}\).
angle addition identity
Angle addition identities are crucial for simplifying trigonometric expressions involving sums or differences of angles. They show how to combine trigonometric functions of two angles into functions of their sum or difference. In our exercise, we're using the angle addition identity for sine:
\(\sin(a + b) = \sin(a) \cos(b) + \cos(a) \sin(b)\).
Since we know:
  • \(\sin\left(\frac{3 \pi}{2}\right) = -1\), and
  • \(\cos\left(\frac{3 \pi}{2}\right) = 0\),

We substitute these values into the addition formula:
  • \(\sin \left( \frac{3 \pi}{2} + \theta \right) = (-1) \cdot \cos(\theta) + 0 \cdot \sin(\theta)\)

Thus simplifying to:
\(\sin \left( \frac{3 \pi}{2} + \theta \right) = -\cos(\theta)\).
trigonometric simplification
Trigonometric simplification involves reducing complex trigonometric expressions to simpler forms. In our exercise, starting with \(\sin \left( \frac{3 \pi}{2} + \theta \right)\), we used the sine addition formula and known values for sine and cosine at specific angles:
  • \(\sin\left(\frac{3 \pi}{2}\right) = -1\)
  • \(\cos\left(\frac{3 \pi}{2}\right) = 0\)
Simplification steps followed:
  • Express the given trigonometric function using the angle addition identity.
  • Substitute known values of trigonometric functions.
  • Simplify the resulting expression.
So, starting from \(\sin \left( \frac{3 \pi}{2} + \theta \right)\), we applied the identity and substituted values to reach: \( -\cos(\theta)\). This process demonstrates breaking down a complex problem into manageable steps to reach a simplified solution.

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