Problem 10 - Algebra 2 and Trigonometry

Question

In \(3-22 :\) a. Rewrite each function value in terms of its cofunction. b. Find, to four decimal places, the value of the function value found in a. $$ \cos 70^{\circ} $$

Step-by-Step Solution

Answer

\( \cos 70^{\circ} = \sin 20^{\circ} \approx 0.3420 \).

Step 1: Understanding Cofunction Identities

Cofunction identities are trigonometric identities that relate a trigonometric function of an angle to the cofunction of its complement. Specifically, for cosine and sine, the identity is \( \cos(\theta) = \sin(90^\circ - \theta) \).

Step 2: Rewrite Cosine in Terms of Cofunction

Using the cofunction identity \( \cos(\theta) = \sin(90^\circ - \theta) \), we can rewrite \( \cos 70^{\circ} \) as a sine function: \( \cos 70^{\circ} = \sin(90^{\circ} - 70^{\circ}) \). Simplifying, we get \( \sin 20^{\circ} \).

Step 3: Calculate the Sine Value

Now that we have \( \sin 20^{\circ} \), use a calculator to find its numerical value. To four decimal places, \( \sin 20^{\circ} \approx 0.3420 \).

Key Concepts

Cofunction Identities

Cofunction identities are powerful tools in trigonometry that help relate the angles of trigonometric functions to one another. They are based on the complementary angle concept, where two angles add up to 90 degrees. For instance, the basic idea is that the trigonometric function of an angle is equal to the cofunction of its complement.

• The cofunction identity for cosine and sine is: \( \cos(\theta) = \sin(90^\circ - \theta) \)
• This principle can also be extended to other pairs, such as sine and cosine, tangent and cotangent, secant and cosecant.

These identities are incredibly useful when solving trigonometric equations, simplifying expressions, or evaluating specific angle values. The ability to switch between functions through these identities provides flexibility and simplifies many complex mathematical problems.

Trigonometric Functions

Trigonometric functions, primarily the sine, cosine, and tangent functions, are fundamental in mathematics. They relate the angles of a triangle to the lengths of its sides. Understanding these functions is critical in fields such as physics, engineering, and computer graphics.

• Sine (sin) Function: Relates the opposite side to the hypotenuse in a right triangle.
• Cosine (cos) Function: Relates the adjacent side to the hypotenuse.
• Tangent (tan) Function: Relates the opposite side to the adjacent side.

There are three additional trigonometric functions, but the primary focus is often on these three for foundational understanding. Each function provides unique insights and is used based on the contextual need within a problem. For instance, cosine and sine are closely connected through their cofunction relationships.

Cosine and Sine Relationship

The relationship between cosine and sine is a profound concept stemming from cofunction identities. As previously covered, one of the main expressions of their relationship is derived from complementary angles.

• For any angle \( \theta \), \( \cos(\theta) = \sin(90^\circ - \theta) \)
• This relationship also highlights that cosine and sine are merely phase-shifted versions of each other. Say, if you graph them over angles, their areas of zero and repetition will align perfectly.

Understanding this relationship is vital in more advanced studies such as harmonic motion, wave analysis, and alternating current circuits. It simplifies calculations, as seen in even simple tasks such as rewriting \( \cos 70^\circ \) as \( \sin 20^\circ \). By recognizing these inherent connections, solving trigonometric problems becomes a more intuitive process.