Problem 59 - Algebra and Trigonometry

Question

Based on material learned earlier in the course. The purpose of these problems is to keep the material fresh in your mind so that you are better prepared for the final exam. Find the exact value of \(\cos 80^{\circ} \cos 70^{\circ}-\sin 80^{\circ} \sin 70^{\circ}\).

Step-by-Step Solution

Answer

-\( \frac{\sqrt{3}}{2} \)

Step 1: Identify the Trigonometric Identity

Recall the cosine of the sum of two angles identity: \ \ \[ \cos (A + B) = \cos A \cos B - \sin A \sin B \]

Step 2: Assign Values to the Angles

In this exercise, assign \( A = 80^{\circ} \) and \( B = 70^{\circ} \). Thus: \( \cos 80^{\circ} \cos 70^{\circ} - \sin 80^{\circ} \sin 70^{\circ} = \cos (80^{\circ} + 70^{\circ}) \).

Step 3: Compute the Sum of the Angles

Add \( 80^{\circ} \) and \( 70^{\circ} \): \( 80^{\circ} + 70^{\circ} = 150^{\circ} \).

Step 4: Evaluate the Cosine

Substitute the sum back into the identity: \( \cos (80^{\circ} + 70^{\circ}) = \cos 150^{\circ} \). Now recall the known value: \( \cos 150^{\circ} = -\frac{\sqrt{3}}{2} \).

Key Concepts

cosine sum identity

The cosine sum identity is one of the fundamental trigonometric identities. It helps simplify expressions involving the cosine of the sum of two angles. This identity is written as: \[ \cos (A + B) = \cos A \cos B - \sin A \sin B \]. \ Learning this identity is helpful for solving various trigonometric problems. It essentially demonstrates that we can use the components (cosines and sines) of individual angles to find the cosine of their sum. Whenever you see a problem asking to find an expression like \[ \cos A \cos B - \sin A \sin B \], the cosine sum identity should come to mind immediately.

exact trigonometric values

Trigonometric values for common angles, such as 30°, 45°, 60°, 90°, and their multiples, can be found exactly. This means we can express these values using simple fractions and square roots. For example: \ \[ \cos 0^{\circ} = 1, \cos 30^{\circ} = \frac{\root 3}{2}, \cos 45^{\circ} = \frac{1}{\root 2}, \cos 60^{\circ} = \frac{1}{2}, \cos 90^{\circ} = 0 \]. \ Having these values memorized or easily accessible helps a lot when tackling trigonometric problems. In our example, knowing that \ \[ \cos 150^{\circ} = -\frac{\root 3}{2} \] aided in finding the exact value directly.

angle addition formulas

Angle addition formulas are used to calculate the trigonometric functions of sums or differences of two angles. In our case, we used the cosine addition formula: \ \[ \cos (A + B) = \cos A \cos B - \sin A \sin B \]. \ This formula can be expanded to other functions, such as sine and tangent. For example: \ \[ \sin (A + B) = \sin A \cos B + \cos A \sin B \] \ and \ \[ \tan (A + B) = \frac{\tan A + \tan B}{1 - \tan A \tan B} \]. \ These formulas are particularly useful in trigonometry as they reduce complex calculations into simpler ones.

cosine function evaluation

When evaluating the cosine function for specific angles, especially those not immediately obvious, angle addition formulas become quite useful. For example, here we calculated \ \[ \cos (80^{\circ} + 70^{\circ}) = \cos 150^{\circ} \], and we knew from the exact values that \ \[ \cos 150^{\circ} = -\frac{\root 3}{2} \]. \ Hence, the identity simplified our original expression. \ Breaking down angles into sums or differences of known angles, then using their exact trigonometric values, often simplifies your work considerably. \ Evaluating the cosine function this way is a handy method to have in your mathematical toolbox.