Problem 58 - Algebra 2

Question

Find values of a and \(b\) such that the function \(y=\sin \theta\) can be expressed as \(y=a \cos (\theta+b)\)

Step-by-Step Solution

Answer

The values of a and b that allow us to express the function y = sin(θ) as y = a cos(θ+b) are a = 1 and b = 90°.

Step 1: Recall the identity

Recall the trigonometric identity that states \( \sin (\theta) = \cos (90° - \theta) \). Therefore, for the equation to hold, a must equal 1 and b must be \(90°\), as \(\sin(\theta)\) can be rewritten as \( \cos(90° - \theta) \).

Step 2: Confirmation

Substituting a = 1 and b = 90° into the equation gives \( y = 1 \cos(\theta + 90°) \). We can confirm that \( \sin(\theta) \) and \( 1 \cos(\theta + 90°) \) are equivalent trigonometric expressions, thus validating our findings.

Key Concepts

Sine and Cosine Relationships

The relationship between sine and cosine is one of the fundamental concepts in trigonometry. Sine and cosine are trigonometric functions that describe relationships in a right-angled triangle, especially concerning angles and lengths of sides. One key relationship is the complementary angle identity. This identity states that for any angle \( \theta \), the sine can be expressed as the cosine of its complementary angle:

• \(\sin(\theta) = \cos(90^\circ - \theta)\)

This means that the sine of an angle is equal to the cosine of the angle's complement. This identity is crucial when transforming or adjusting trigonometric functions in various mathematical contexts, such as rewriting expressions or simplifying complex equations. Knowing these relationships helps understand that sine and cosine are essentially the same function, just shifted horizontally relative to one another by \(90^\circ\). This insight will simplify many trigonometry problems as it allows one to exchange sine for cosine and vice versa.

Phase Shift

A phase shift refers to the horizontal displacement of a trigonometric function on a graph. It involves shifting the wave of a function left or right without altering its shape. In trigonometric terms, any expression of form \(y = a \cos(\theta + b)\) indicates a phase shift:

• \(b\) is the phase shift of the function.

When examining the function \(y = \cos(\theta + 90^\circ)\), the phase shift is evident. The graph of \(\cos(\theta)\) moves 90 degrees to the left, aligning perfectly with the graph of \(\sin(\theta)\). Understanding phase shifts is crucial for transforming one trigonometric function into another. This is vital when analyzing wave-like data or solving trigonometric equations. By recognizing the phase shift, you can transform a sine function into a cosine function and vice versa, thus making calculations more convenient.

Trigonometric Equations

Trigonometric equations are equations that involve trigonometric functions like sine, cosine, and tangent. Solving these equations is an essential skill, as they appear frequently in various mathematical applications. To solve an equation like \(y = \sin \theta = a \cos(\theta + b)\), you must match the two functions:

• The identity \(\sin(\theta) = \cos(90^\circ - \theta)\) lets you express \(\sin(\theta)\) in terms of a cosine function.

By understanding the trigonometric properties, you can substitute values appropriately to find solutions. In the original problem, for \(a = 1\) and \(b = 90^\circ\), the equation holds true. This simplification and substitution step is typical in trigonometry. It involves recognizing equivalent trigonometric expressions to solve or simplify equations, making trigonometry an invaluable tool when dealing with periodic or cyclical phenomena in both mathematics and real life.