Problem 55
Question
Use the sum and difference formulas to verify each identity. $$ \sin (\pi+\theta)=-\sin \theta $$
Step-by-Step Solution
Verified Answer
Therefore, the given identity \(\sin(\pi + \theta) = -\sin\theta\) is verified.
1Step 1: Recognize the pattern
The given expression \(\sin(\pi + \theta)\) is represented in the form of \(\sin(A + B)\). This pattern can be recognized as the sum formula in trigonometry.
2Step 2: Apply the Sum Formula
Apply the sum formula: \(\sin(A + B) = \sin A \cos B + \cos A \sin B\). Letting \(A = \pi\) and \(B = \theta\), the equation \(\sin(\pi + \theta)\) becomes \(\sin \pi \cos \theta + \cos \pi \sin \theta\).
3Step 3: Evaluate the values
From trigonometric exact values, we know that \(\sin \pi = 0\) and \(\cos \pi = -1\). Substituting these values, we have \(0 \cdot \cos \theta - 1 \cdot \sin \theta = -\sin \theta\).
Key Concepts
Sum and Difference FormulasSine FunctionAngle Addition Formulas
Sum and Difference Formulas
The sum and difference formulas in trigonometry are powerful tools that allow us to express trigonometric functions of sums or differences of angles in terms of the functions of the individual angles. These formulas are particularly useful in simplifying complex trigonometric expressions and solving equations.
\[ \begin{align*} \sin(A + B) & = \sin A \cos B + \cos A \sin B \\sin(A - B) & = \sin A \cos B - \cos A \sin B \\cos(A + B) & = \cos A \cos B - \sin A \sin B \\cos(A - B) & = \cos A \cos B + \sin A \sin B \end{align*} \]
In this exercise, we use the sum formula for sine, which helps express the sine of a sum of two angles, \(A\) and \(B\), as a sum of products involving the sine and cosine of those angles. Understanding these formulas allows us to verify the given identity \(\sin (\pi+\theta)=-\sin \theta\).
\[ \begin{align*} \sin(A + B) & = \sin A \cos B + \cos A \sin B \\sin(A - B) & = \sin A \cos B - \cos A \sin B \\cos(A + B) & = \cos A \cos B - \sin A \sin B \\cos(A - B) & = \cos A \cos B + \sin A \sin B \end{align*} \]
In this exercise, we use the sum formula for sine, which helps express the sine of a sum of two angles, \(A\) and \(B\), as a sum of products involving the sine and cosine of those angles. Understanding these formulas allows us to verify the given identity \(\sin (\pi+\theta)=-\sin \theta\).
Sine Function
The sine function is one of the most fundamental trigonometric functions. It relates the angle in a right triangle to the ratio of the length of the side opposite the angle to the hypotenuse.
When working with the exercise's identity, \(\sin(\pi + \theta)\), knowing the properties of the sine function, especially its periodic nature and symmetry, helps us simplify and validate the identity. The sine wave is symmetric around its origin and reaches its maximum amplitude of 1 at \(\frac{\pi}{2}\) radians, making it essential for solving various trigonometric problems.
- The sine of an angle \(\theta\) is denoted as \(\sin \theta\).
- For angles measured in radians, \(\sin \theta\) ranges between -1 and 1.
- Important trigonometric identities involving sine include \(\sin^2 \theta + \cos^2 \theta = 1\).
When working with the exercise's identity, \(\sin(\pi + \theta)\), knowing the properties of the sine function, especially its periodic nature and symmetry, helps us simplify and validate the identity. The sine wave is symmetric around its origin and reaches its maximum amplitude of 1 at \(\frac{\pi}{2}\) radians, making it essential for solving various trigonometric problems.
Angle Addition Formulas
Angle addition formulas play a crucial role in trigonometry by enabling us to calculate the trigonometric functions of the sum of two angles. This is particularly useful in many real-world applications and in various fields of science and engineering.
The sine angle addition formula is:
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]This formula allows us to break down a complex angle expression into simpler components, which can be individually evaluated and easily combined. In the given exercise, we explored \(\sin(\pi + \theta)\), where \(A = \pi\) and \(B = \theta\), employing the angle addition formula to step through the identity verification. Knowing that \(\sin \pi = 0\) and \(\cos \pi = -1\), we see how the formula simplifies to \(-\sin \theta\), demonstrating the formula's utility in verifying trigonometric identities.
The sine angle addition formula is:
\[ \sin(A + B) = \sin A \cos B + \cos A \sin B \]This formula allows us to break down a complex angle expression into simpler components, which can be individually evaluated and easily combined. In the given exercise, we explored \(\sin(\pi + \theta)\), where \(A = \pi\) and \(B = \theta\), employing the angle addition formula to step through the identity verification. Knowing that \(\sin \pi = 0\) and \(\cos \pi = -1\), we see how the formula simplifies to \(-\sin \theta\), demonstrating the formula's utility in verifying trigonometric identities.
Other exercises in this chapter
Problem 55
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Find each angle measure to the nearest tenth of a degree. $$ \tan ^{-1} \sqrt{2} $$
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