Problem 44 - A Graphical Approach to Precalculus with Limits

Question

Write each expression in terms of its co-function. $$\sin \frac{\pi}{3}$$

Step-by-Step Solution

Answer

\(\sin \frac{\pi}{3} = \cos \frac{\pi}{6}\)

Step 1: Identify the Co-function Formula

Recall the co-function identity for sine: \(\sin(\theta) = \cos\left(\frac{\pi}{2} - \theta\right)\). This identity relates sine to cosine through the complementary angle.

Step 2: Apply the Angle to the Co-function Formula

Given \(\theta = \frac{\pi}{3}\), substitute it into the co-function identity: \(\sin\left(\frac{\pi}{3}\right) = \cos\left(\frac{\pi}{2} - \frac{\pi}{3}\right)\).

Step 3: Simplify the Expression

Calculate the complementary angle: \(\frac{\pi}{2} - \frac{\pi}{3} = \frac{3\pi}{6} - \frac{2\pi}{6} = \frac{\pi}{6}\).

Step 4: Write the Expression in Terms of the Co-function

Substitute back into the identity with the simplified angle: \(\sin\left(\frac{\pi}{3}\right) = \cos\left(\frac{\pi}{6}\right)\). Thus, the expression is now in terms of its co-function, cosine.

Key Concepts

Co-functions

Co-functions are pairs of trigonometric functions that are complementary. This means they describe the relationship between angles and their complementary parts. Complementary angles are two angles that add up to a right angle or \[\frac{\pi}{2} \] in radians. This is a crucial concept for co-functions. In simple terms, one function can be expressed in terms of another by using this relationship.

The most common co-function pairs are:

• Sine and Cosine
• Tangent and Cotangent
• Secant and Cosecant

In the context of this exercise, we are focusing on sine and cosine. If you know the value of sine for a certain angle, you can find the value of cosine for its complementary angle, and vice versa. This is beautifully captured by the identity: \[\sin(\theta) = \cos\left(\frac{\pi}{2} - \theta\right) \] Understanding co-function identities simplifies the process of solving trigonometric expressions, transforming them into an easier or more familiar form.

Sine and Cosine

Sine and cosine are two primary trigonometric functions. They are often the first pair of functions learned in trigonometry.

**Sine Function:** It represents the y-coordinate of a point on the unit circle. If you imagine a circle with a radius of 1, sine gives the vertical projection of the point.**Cosine Function:** Conversely, cosine represents the x-coordinate of that same point on the unit circle. It gives us the horizontal projection.The intimate connection between sine and cosine arises from their roles on the unit circle and their complementary relationship.

• They are co-functions of each other, meaning the sine of an angle is the cosine of its complement.
• They oscillate between -1 and 1 as the angle moves around the circle.

In an exercise like this one, using the identity: \[\sin\left(\frac{\pi}{3}\right) = \cos\left(\frac{\pi}{6}\right)\] the solution uses the unit circle properties to express one function in terms of the other.

Angle Measurement in Radians

In mathematics, angles are typically measured in radians. Radians provide a natural way of measuring angles in terms of the arc length they subtend on a circle.

Unlike degrees, which divide a circle into 360 parts, radians relate directly to the circumference of a circle. One radian equates to the angle created when the arc length is equal to the radius of the circle. Thus, a full circle is \[2\pi\] radians.

• \[\frac{\pi}{2} \]radians represents a quarter turn of the circle (90 degrees).
• \[\frac{\pi}{3} \] and\[\frac{\pi}{6} \]are common angles often encountered in trigonometry.

Using radians simplifies the application of trigonometric identities. In this problem, using radians allows us to directly apply the co-function identities:The conversion between radians and degrees is an essential skill, but for many trigonometric identities, radians provide a more intuitive framework to work within.