Problem 14 - Algebra and Trigonometry

Question

\(13-18\) m Use an addition or subtraction formula to write the expression as a trigonometric function of one number, and then find its exact value. \(\cos 10^{\circ} \cos 80^{\circ}-\sin 10^{\circ} \sin 80^{\circ}\)

Step-by-Step Solution

Answer

The exact value is 0.

Step 1: Identify the Sine Subtraction Formula

The expression given, \( \cos 10^{\circ} \cos 80^{\circ} - \sin 10^{\circ} \sin 80^{\circ} \), matches the cosine subtraction identity: \( \cos(a)\cos(b) - \sin(a)\sin(b) = \cos(a+b) \). The angle of subtraction in this identity is \( a + b \).

Step 2: Apply the Formula

Given the cosine subtraction identity, substitute \( a = 10^{\circ} \) and \( b = 80^{\circ} \). This gives us: \[ \cos(10^{\circ} + 80^{\circ}) = \cos(90^{\circ}) \].

Step 3: Calculate Cosine of 90 Degrees

Determine the value of \( \cos(90^{\circ}) \), which is 0. Cosine of 90 degrees is zero because, on the unit circle, the x-coordinate (cosine value) of 90 degrees is 0.

Key Concepts

Cosine Subtraction Formula

Trigonometric identities are mathematical expressions that hold true for all values of the variables involved. One important identity is the cosine subtraction formula. This formula allows us to simplify expressions involving the cosine and sine of two angles. Specifically, the formula is:

• \( \cos(a) \cos(b) - \sin(a) \sin(b) = \cos(a+b) \)

This formula is useful because it helps us condense two trigonometric expressions into one.
By recognizing patterns in equations, we can rewrite them in a simpler form. In the problem you provided, the task is to use this identity to combine two cosine and sine products into a single cosine term. This is an essential skill for solving trigonometric equations efficiently.

Unit Circle

To fully understand trigonometric identities, it is vital to grasp the concept of the unit circle. The unit circle is a circle with a radius of 1 centered at the origin of a coordinate system. The coordinates of any point on this circle directly relate to the sine and cosine of the angle formed by the radius and the positive x-axis.

• The x-coordinate represents the cosine of the angle.
• The y-coordinate represents the sine of the angle.

This geometric representation provides an intuitive way to understand angle measures and their corresponding trigonometric values. For example, when we say the cosine of 90 degrees is zero, it means that at 90 degrees, the point on the unit circle lies directly on the y-axis, where the x-coordinate is indeed 0.

Exact Trigonometric Values

Trigonometry often involves finding exact values for certain angles. These exact values are crucial to solving equations accurately and efficiently.
The term 'exact values' refers to the specific sine or cosine values of common angles such as 0°, 30°, 45°, 60°, and 90° without an approximation.
Some noteworthy exact trigonometric values include:

• \( \cos(0°) = 1 \)
• \( \cos(30°) = \sqrt{3}/2 \)
• \( \cos(45°) = \sqrt{2}/2 \)
• \( \cos(60°) = 1/2 \)
• \( \cos(90°) = 0 \)

Similarly for the sine function:

• \( \sin(0°) = 0 \)
• \( \sin(30°) = 1/2 \)
• \( \sin(45°) = \sqrt{2}/2 \)
• \( \sin(60°) = \sqrt{3}/2 \)
• \( \sin(90°) = 1 \)

Understanding these values is key to simplifying trigonometric expressions and solving related problems, including those involving subtraction or addition formulas.