Problem 26 - Algebra and Trigonometry

Question

Use one or more of the six sum and difference identities to solve Exercises \(13-54\) Write each expression as the sine, cosine, or tangent of an angle. Then find the exact value of the expression. $$ \sin 40^{\circ} \cos 20^{\circ}+\cos 40^{\circ} \sin 20^{\circ} $$

Step-by-Step Solution

Answer

The exact value of the expression is \( \frac{\sqrt{3}}{2} \)

Step 1: Identity Application

Rewrite \( \sin 40^{\circ} \cos 20^{\circ}+\cos 40^{\circ} \sin 20^{\circ} \) as \( \sin (40^{\circ} + 20^{\circ}) \) using the identity \( \sin(A + B) = \sin A \cos B+ \cos A \sin B \)

Step 2: Simplify the Expression

Simplify the expression \( \sin (40^{\circ} + 20^{\circ}) \) to \( \sin 60^{\circ} \)

Step 3: Find Exact Value

The exact value of \( \sin 60^{\circ} \) is \( \frac{\sqrt{3}}{2} \)

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is crucial when attempting to simplify trigonometric expressions or solve equations involving trigonometric functions. These identities are mathematical statements that establish the relationship between various trigonometric functions. One commonly used set of identities is the sum and difference identities for sine, cosine, and tangent.

For our example, we used a sum identity for sine: \[ \text{sin}(A + B) = \text{sin} A \text{cos} B + \text{cos} A \text{sin} B \] This identity was the key in transforming the original expression \( \text{sin} 40^\circ \text{cos} 20^\circ + \text{cos} 40^\circ \text{sin} 20^\circ \) into the simplified form \(\text{sin}(60^\circ)\).

Being fluent in such identities allows you to approach and solve a wide array of trigonometry problems, ranging from those related to angles and geometric figures to complex real-world applications. Memorization paired with regular practice helps in recognizing when and how to apply these identities effectively.

Sine of an Angle

The sine of an angle is one of the basic trigonometric functions. In a right-angled triangle, the sine of an angle is defined as the ratio of the length of the opposite side to the hypotenuse. In the context of the unit circle, it's the y-coordinate of a point where a line through the angle intersects the circle.

In our example, we simplified an expression using sine to find the exact value of \( \text{sin} 60^\circ \) which is \(\frac{\text{sqrt}{3}}{2}\). This value is derived from the properties of an equilateral triangle, where each angle measures 60 degrees. Knowing these exact values for specific angles—like 30°, 45°, and 60°—is incredibly useful in solving trigonometry problems quickly and accurately.

Cosine of an Angle

The cosine function relates to the sine function but instead uses the adjacent side and the hypotenuse in a right-angled triangle. It can also be visualized on the unit circle, where it represents the x-coordinate of the point where the line intersects the circle. Just like the sine function, certain angles have commonly known cosine values.

Understanding cosines is essential as they often appear in conjunction with sine, especially in sum and difference identities. For instance, to arrive at the solution in our exercise, we used the known values and properties of both the sine and cosine functions of the angles in question.

Tangent of an Angle

Lastly, the tangent of an angle is another fundamental trigonometric function. The tangent is defined as the ratio of the sine to the cosine of that angle. In the context of a right-angled triangle, it is the ratio of the opposite side to the adjacent side. On the unit circle, it's associated with the slopes of lines intersecting it.

The tangent function isn't directly used in the problem at hand, but it's worth noting that it has its own sum and difference identities. These identities can make solving certain trigonometry problems much more direct and are equally vital as those for sine and cosine.