Problem 4 - 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. $$ \cos 80^{\circ} $$

Step-by-Step Solution

Answer

a. \(\cos 80^{\circ} = \sin 10^{\circ}\); b. \(\sin 10^{\circ} \approx 0.1736\).

Step 1: Understanding Co-Function Identity

The co-function identity states that the cosine of an angle is equal to the sine of its complement. Therefore, we can express \(\cos \theta\) in terms of sine using the identity: \(\cos \theta = \sin(90^{\circ} - \theta)\).

Step 2: Applying the Co-Function Identity

Apply the co-function identity to \(\cos 80^{\circ}\). We find its complement of 80 degrees by subtracting 80 from 90: \(90^{\circ} - 80^{\circ} = 10^{\circ}\). Thus, \(\cos 80^{\circ} = \sin 10^{\circ}\).

Step 3: Calculating the Sine Value

We need to find the numerical value of \(\sin 10^{\circ}\). Using a calculator, input 10 and find the sine. Ensure the calculator is in degree mode. The value of \(\sin 10^{\circ}\) is approximately 0.1736 when rounded to four decimal places.

Key Concepts

Cosine

The cosine function is a fundamental part of trigonometry and is used to relate the sides of a right triangle to the angles in the triangle. When dealing with angles, cosine specifically works as a ratio comparing the length of the adjacent side to the hypotenuse.
For example, in a right triangle, if an angle \( \theta \) has an adjacent side of length 'a' and a hypotenuse of length 'c', then the cosine of \( \theta \) is given by the formula:

• \( \ \cos \theta = \frac{a}{c} \ \)

Cosine is crucial not only in geometry but also in fields like physics and engineering, where it helps to understand wave patterns and forces.

Sine

The sine function, another essential part of trigonometry, also deals with the relationships between the sides of a right triangle and its angles. Unlike cosine, the sine function represents the ratio of the opposite side to the hypotenuse.
Using the notation for a right triangle, if \( \theta \) is an angle with an opposite side of length 'o' and a hypotenuse of 'h', the sine of \( \theta \) is calculated as:

• \( \ \sin \theta = \frac{o}{h} \ \)

Sine is used in various applications, particularly in the study of oscillations and sound waves, where it can help model periodic phenomena.

Trigonometry

Trigonometry is a branch of mathematics focused on studying triangles, particularly right triangles, and the relationships between their angles and sides. It's foundational for fields such as physics, engineering, and computer graphics.
Key functions in trigonometry include sine (

• cosine
• sine
• \( \ tangent \ \)

), each providing unique insights into different geometric relationships.
Trigonometry goes beyond basic triangles to analyze waves and oscillations, predict astrophysical phenomena, and more. Its usefulness extends through scientific disciplines and everyday applications, like building and constructing, where angle measurement is crucial.

Angle Complements

In trigonometry, the idea of complementary angles plays an integral role. Complementary angles are two angles whose sum is 90 degrees, leading to their particular interest in right triangle analysis.
When using cofunction identities, these complementary angles become significant. The primary identity shows that

• \( \ \cos \theta = \sin(90^{\circ} - \theta) \)

which directly relates the cosine of one angle to the sine of its complementary angle.
Understanding complementary angles is crucial for solving various trigonometric problems, such as identifying unknown angles or rewriting functions in terms of their complements.