Question

Derive the cofunction identity \(\sin \left(\frac{\pi}{2}-\theta\right)=\cos \theta\) using the difference formula for sine.

Short answer

By applying the difference formula for sine and substituting the known trigonometric values, it is possible to derive the cofunction identity \(\sin \left(\frac{\pi}{2}-\theta\right)=\cos \theta\).

Step-by-step solution

Apply the Difference Formula

Start by replacing \(a\) with \(\frac{\pi}{2}\) and \(b\) with \(\theta\) in the difference formula. This gives you \(\sin \left(\frac{\pi}{2}-\theta\right) = \sin \frac{\pi}{2} \cos \theta - \cos \frac{\pi}{2} \sin \theta\).

Substitute Known Trigonometric Values

Next, substitute the known values of \(\sin \frac{\pi}{2}\) which is 1, and \(\cos\frac{\pi}{2}\) which is 0, into the equation. This will simplify the expression to \(\sin \left(\frac{\pi}{2}-\theta\right) = 1 \times \cos \theta - 0 \times \sin \theta\).

Simplify the Expression

Now, simplify the expression further. This essentially reduces to \(\sin \left(\frac{\pi}{2}-\theta\right) = \cos \theta\).

Key concepts

Difference Formula for Sine

Understanding the difference formula for sine is crucial when solving trigonometric problems involving the subtraction of angles. It is one of the fundamental tools in trigonometry used to calculate the sine of the difference between any two angles.

Mathematically, the formula is represented as:
\[ \sin(a - b) = \sin(a)\cos(b) - \cos(a)\sin(b) \]
Where \( a \) and \( b \) are angle measures. This formula allows us to express the sine of a difference of angles in terms of the sine and cosine of individual angles, which often makes calculations simpler as we can then use known values of sine and cosine for specific angle measures to solve problems.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all values of the included variables. They play an indispensable role in simplifying trigonometric expressions and proving other mathematical theorems. Some of the most commonly used trigonometric identities include the Pythagorean identities, angle sum and difference identities, double angle identities, and cofunction identities.

For example, the Pythagorean identity connects the sine and cosine of an angle in a straightforward way:\[ \sin^2(\theta) + \cos^2(\theta) = 1 \]
Understanding these identities is vital not only in trigonometry but also in calculus and other areas of mathematics where trigonometric functions are applied. Memorable identities like \( \sin \left( \frac{\pi}{2} \right) = 1 \) and \( \cos \left( \frac{\pi}{2} \right) = 0 \) help simplify many trigonometric expressions.

Simplifying Trigonometric Expressions

When simplifying trigonometric expressions, it's essential to utilize the identities and formulas you know. This process often includes acknowledging the specific values of trigonometric functions at special angles, applying the relevant identities accurately, and algebraically manipulating the expression to its simplest form.

Approach simplification step by step:

• Identify and apply pertinent trigonometric identities.
• Substitute known values for trigonometric functions at special angles.
• Perform algebraic operations to condense the expression.
• Cancel common factors if present.

By doing so, you can translate complicated trigonometric expressions into more manageable terms or, at times, a single function, as seen in the cofunction identity \( \sin \left( \frac{\pi}{2} - \theta \right) = \cos(\theta) \). This identity itself emerges from simplifying the difference formula for sine by substituting specific angle values and simplifying the result.