Problem 48 - A Graphical Approach to Precalculus with Limits

Question

Write each expression in terms of its co-function. $$\sin \frac{1}{2}$$

Step-by-Step Solution

Answer

\( \sin\left(\frac{1}{2}\right) = \cos\left(\frac{\pi - 1}{2}\right) \)

Step 1: Understand Co-Function Identities

Co-function identities relate trigonometric functions of complementary angles. Two angles are complementary if their sum is \(90^\circ\) or \(\frac{\pi}{2}\) radians. The important identities for sine and cosine are: \( \sin(\theta) = \cos(\frac{\pi}{2} - \theta) \) and \( \cos(\theta) = \sin(\frac{\pi}{2} - \theta) \). This means each trigonometric function can be rewritten in terms of its co-function using this identity.

Step 2: Identify the Given Expression

The exercise gives the trigonometric expression \( \sin\left(\frac{1}{2}\right) \). Here, \( \frac{1}{2} \) is the angle in radians that we want to express in terms of its co-function.

Step 3: Apply the Co-Function Identity for Sine

Using the co-function identity for sine, \( \sin(\frac{1}{2}) = \cos(\frac{\pi}{2} - \frac{1}{2}) \). Calculate \( \frac{\pi}{2} - \frac{1}{2} \) to find the angle in terms of cosine.

Step 4: Calculate \( \frac{\pi}{2} - \frac{1}{2} \)

Perform the subtraction: \[ \frac{\pi}{2} - \frac{1}{2} = \frac{\pi - 1}{2} \]. Substitute this result back into the co-function expression.

Step 5: Write the Final Expression

Now we can express \( \sin\left(\frac{1}{2}\right) \) in terms of its co-function: \( \sin\left(\frac{1}{2}\right) = \cos\left(\frac{\pi - 1}{2}\right) \). This is the expression written in terms of its co-function.

Key Concepts

Trigonometric Functions

Trigonometric functions are fundamental in mathematics, especially in the study of triangles and modeling periodic phenomena. These functions include sine, cosine, tangent, and their reciprocals: cosecant, secant, and cotangent.

• Sine (\(\sin\theta\)) measures the ratio of the length of the side opposite the angle to the hypotenuse in a right-angle triangle.
• Cosine (\(\cos\theta\)) represents the ratio of the adjacent side to the hypotenuse.
• Tangent (\(\tan\theta\)) is the ratio of the opposite side to the adjacent side.

Trigonometric functions are periodic and repeat values in a regular pattern over specific intervals. Understanding these concepts is essential, as they form the basis for more complex topics like co-function identities, which are used to relate the values of sine and cosine.

Complementary Angles

Complementary angles are two angles that add up to a right angle (\(90^\circ\)) or \(\frac{\pi}{2}\) radians. This concept is crucial when exploring co-function identities in trigonometry.
Co-function identities leverage complementary angles to express any trigonometric function, such as sine or cosine, via its co-function.

• Sine and Cosine: \(\sin\theta = \cos(\frac{\pi}{2} - \theta)\)
• Cosine and Sine: \(\cos\theta = \sin(\frac{\pi}{2} - \theta)\)

These relationships show how understanding one function can provide insight into its complementary co-function, simplifying mathematical transformations and problem-solving in trigonometry.

Radians

Radians offer a different perspective compared to degrees, allowing a natural approach for measuring angles based on the circumference of a circle.
The radian measure relates directly to arc length, where one radian is the angle formed when the arc length equals the radius of the circle.
There are \(2\pi\) radians in a complete circle, analogous to \(360^\circ\).

• A right angle, or \(90^\circ\), is equivalent to \(\frac{\pi}{2}\) radians.
• One radian is approximately \(57.2958^\circ\)

Radians are often preferred in calculus and advanced mathematics due to their convenient properties when used in derivatives and integrals, making them an essential concept in mastering trigonometry.