Problem 9 - Algebra 2

Question

Verify that each of the following is an identity. \(\sin \left(\theta+\frac{\pi}{2}\right)=\cos \theta\)

Step-by-Step Solution

Answer

The identity is verified: \( \sin \left(\theta + \frac{\pi}{2}\right) = \cos \theta \).

Step 1: Understand the problem

We need to verify the trigonometric identity \(\sin \left(\theta+\frac{\pi}{2}\right)=\cos \theta\). This means showing that both sides of the equation are equal for all values of \(\theta\).

Step 2: Use the Sine Addition Formula

Recall the sine addition formula: \( \sin(a + b) = \sin a \cos b + \cos a \sin b \). We can apply this formula with \( a = \theta \) and \( b = \frac{\pi}{2} \).

Step 3: Substitute into the Sine Addition Formula

Substitute \( a = \theta \) and \( b = \frac{\pi}{2} \) into the formula: \[ \sin \left(\theta + \frac{\pi}{2}\right) = \sin \theta \cos \frac{\pi}{2} + \cos \theta \sin \frac{\pi}{2} \]

Step 4: Evaluate Trigonometric Values

We know from trigonometric identities that \( \cos \frac{\pi}{2} = 0 \) and \( \sin \frac{\pi}{2} = 1 \). Substitute these values into the equation: \[ \sin \left(\theta + \frac{\pi}{2}\right) = \sin \theta \cdot 0 + \cos \theta \cdot 1 \]

Step 5: Simplify the Expression

Simplify the expression: \( \sin \left(\theta + \frac{\pi}{2}\right) = 0 + \cos \theta = \cos \theta \).

Step 6: Conclusion

Since \( \sin \left(\theta + \frac{\pi}{2}\right) = \cos \theta \), we have verified that the given equation is indeed an identity.

Key Concepts

Trigonometric Functions

Trigonometric functions are at the heart of trigonometry, a branch of mathematics that deals with angles and their relationships with circles. These functions help us describe the relationships between the angles and sides of triangles. The most commonly used trigonometric functions are sine (\( \sin \)), cosine (\( \cos \)), and tangent (\( \tan \)). Each function provides a different measure of angles using different sides of the triangle.

• Sine (\( \sin \)) relates the opposite side of a right triangle to its hypotenuse.
• Cosine (\( \cos \)) connects the adjacent side to the hypotenuse.
• Tangent (\( \tan \)) is the ratio of the opposite side to the adjacent side.

Understanding the behaviors of these functions is crucial for verifying trigonometric identities, such as demonstrating that \( \sin \left(\theta+\frac{\pi}{2}\right)=\cos \theta \). By using these relationships, we can manipulate expressions and verify the identities that play significant roles in geometry, physics, engineering, and computer science.

Sine Addition Formula

The sine addition formula is a useful tool that helps us find the sine of two added angles. This formula states that for any angles \(a\) and \(b\), the sine of their sum is given by:
\[ \sin(a + b) = \sin a \cos b + \cos a \sin b \]
This formula allows us to break down complex trigonometric equations into simpler components. In the identity verification exercise, we applied this formula to \( \sin(\theta + \frac{\pi}{2}) \) by setting \( a = \theta \) and \( b = \frac{\pi}{2} \). This substitution gives us a way to express the sine of an angle sum in terms of basic sine and cosine values.
Using the sine addition formula effectively requires practice but it's an invaluable tool in solving trigonometric problems. It allows us to transform and simplify equations, making verification tasks more manageable by expressing sine expressions in terms of other known trigonometric values.

Trigonometric Values Evaluation

Evaluating trigonometric values is an essential part of working with trigonometric identities. Knowledge of these values allows us to simplify and verify equations efficiently. In our example, evaluating specific trigonometric values was crucial:

• Cosine of \( \frac{\pi}{2} \): This value is 0, which is consistent across all applications.
• Sine of \( \frac{\pi}{2} \): This value is always 1, a key fact that simplifies our calculations.

When we substituted these values into our equation, they allowed us to simplify \( \sin(\theta + \frac{\pi}{2}) \) to show it equals \( \cos \theta \). A good grasp of trigonometric value evaluation enables us to quickly determine the relationships within trigonometric identities, serve as proofs, or solve equations accurately. Practice with these values builds intuition for quick recall and application in a wide array of mathematical situations.