Problem 118 - Algebra and Trigonometry

Question

Will help you prepare for the material covered in the next section. In each exercise, use exact values of trigonometric functions to show that the statement is true. Notice that each statement expresses the product of sines and/or cosines as a sum or a difference. $$ \sin \pi \cos \frac{\pi}{2}=\frac{1}{2}\left[\sin \left(\pi+\frac{\pi}{2}\right)+\sin \left(\pi-\frac{\pi}{2}\right)\right] $$

Step-by-Step Solution

Answer

As both sides of the equation evaluate to 0, the statement is true.

Step 1: Evaluate the left side of the equation

Evaluate \(\sin \pi \cos \frac{\pi}{2}\). The \(\sin \pi\) is 0 because the sine of any multiple of \(\pi\) is 0. Remember that cosine of \(\pi/2\) is also 0, as the cosine of \(\pi/2\) plus any integer multiple of \(\pi\) is 0. So, \(\sin \pi \cos \frac{\pi}{2} = 0 * 0 = 0\).

Step 2: Evaluate the right side of the equation

Evaluate \(\frac{1}{2}\left[\sin\left(\pi+\frac{\pi}{2}\right)+\sin \left(\pi-\frac{\pi}{2}\right)\right]\). The \(\sin(\pi + \pi/2)\) equals \(\sin(3\pi/2)\) which equals -1 and \(\sin(\pi - \pi/2)\) equals \(\sin(\pi/2)\) which is 1, so the expression on the right side becomes \(\frac{1}{2} * (-1 + 1) = 0\).

Step 3: Compare both sides of the equation

Comparing both results: the left side of our equation equals to 0 and the right side of our equation also equals to 0, therefore the equation is true, as both sides are equal.

Key Concepts

Sine and Cosine Functions

Sine and cosine are fundamental trigonometric functions that capture essential properties of angles in a unit circle. The sine function, \( \sin(\theta) \), represents the y-coordinate of a point on the circle, whereas the cosine function, \( \cos(\theta) \), corresponds to the x-coordinate. In simpler terms,

• Sine measures the vertical "height" from the circle's horizontal center.
• Cosine measures the horizontal "width" from the circle's vertical center.

These functions oscillate between -1 and 1 as angles vary. Key characteristics include periodicity, with \( \sin(\theta) \) and \( \cos(\theta) \) repeating every \( 2\pi \) radians. They also have symmetrical properties like \( \sin(-\theta) = -\sin(\theta) \) for sine and \( \cos(-\theta) = \cos(\theta) \) for cosine, which reflect their even and odd nature respectively.

Sum and Difference Formulas

The sum and difference formulas for sine and cosine are pivotal for simplifying and evaluating trigonometric expressions. These formulas let us find the trigonometric functions of sums or differences of angles. For sine, it is expressed as:
\[ \sin(a \pm b) = \sin(a)\cos(b) \pm \cos(a)\sin(b) \]
For cosine, the formula is:
\[ \cos(a \pm b) = \cos(a)\cos(b) \mp \sin(a)\sin(b) \]
Notice how the sign before the sine terms in the formula changes with the operation applied in the angles. These formulas not only help in simplifying expressions but also in proving trigonometric identities, as seen in problems where angle expressions are broken down into simpler components that are easier to manage.

Exact Values of Trigonometric Functions

Knowing the exact values of trigonometric functions at specific angles is essential in trigonometry. Exact values are particularly useful at special angles such as \( 0, \pi/6, \pi/4, \pi/3, \pi/2, \ \) and their multiples. For example:

• \( \sin(\pi/6) = \frac{1}{2} \)
• \( \cos(\pi/3) = \frac{1}{2} \)
• \( \sin(\pi) = 0 \)
• \( \cos(\pi/2) = 0 \)

Using these exact values allows us to solve and verify trigonometric equations without relying on calculator approximations. By understanding these values, like how \( \sin(3\pi/2) = -1 \) and \( \sin(\pi/2) = 1 \, \) students can establish the truth of trigonometric identities with certainty, further fostering a stronger grasp of trigonometric principles.