Problem 62 - Precalculus Essentials

Question

Find the exact value of the following under the given conditions: (A) .\(\cos (\alpha+\beta)\) (B). \(\sin (\alpha+\beta)\) (C) \(\tan (\alpha+\beta)\). \(/cos \alpha=\frac{1}{2}, \alpha\) lies in quadrant IV, and \(\sin \beta=-\frac{1}{3}, \beta\) lies in quadrant III.

Step-by-Step Solution

Answer

The exact values of the trigonometric operations are as follows: (A) cos(α+β) = -3√3/6, (B) sin(α+β) = (√24 - 6)/6, and (C) tan(α+β) = -(√24 - 6)/3√3.

Step 1: Evaluate sinα and cosβ

Since cosα = 1/2 and α is in QIV, sinα = - \(\sqrt{1 - cos^2 \alpha} = -\sqrt{1 - (1/2)^2} = -\sqrt{3}/2\). Similarly, since sinβ = -1/3 and β is in QIII, cosβ = -\(\sqrt{1 - sin^2 \beta} = -\sqrt{1 - (-1/3)^2} = -\sqrt{8}/3.\)

Step 2: Evaluate sin(α+β) and cos(α+β)

Using the trigonometric identities for sin(α+β) = sinαcosβ + cosαsinβ = (-√3/2)(-√8/3) + (1/2)(-1/3) = √24/6 - 1/6 = √24/6 - 6/6 = (√24 - 6)/6 and cos(α+β) = cosαcosβ - sinαsinβ = (1/2)(-√8/3) - (-√3/2)(-1/3) = -√4/3 - √3/6 = -2√3/6 - √3/6 = -3√3/6.

Step 3: Evaluate tan(α+β)

Using the trigonometric identity tan(α+β) = sin(α+β)/cos(α+β), substitute the values found in step 2. So, tan(α+β) = [(√24 - 6)/6]/[-3√3/6]= -(√24 - 6)/3√3. Simplify the expression to get the final answer.

Key Concepts

Sum of Angles Formula

Understanding the sum of angles formula in trigonometry is fundamental when dealing with problems involving the addition or subtraction of two angles. It's a tool that allows us to express the sine, cosine, or tangent of the sum (or difference) of two angles in terms of the sines and cosines of the individual angles.

The general formulas for the sum of two angles \( \alpha \) and \( \beta \) are as follows:

• \(\sin(\alpha + \beta) = \sin\alpha\cos\beta + \cos\alpha\sin\beta\)
• \(\cos(\alpha + \beta) = \cos\alpha\cos\beta - \sin\alpha\sin\beta\)
• \(\tan(\alpha + \beta) = \frac{\sin(\alpha + \beta)}{\cos(\alpha + \beta)}\)

These formulas are essential for our exercise as we use them to find the exact trigonometric values for \( \cos(\alpha + \beta) \) and \( \sin(\alpha + \beta) \) by knowing \( \cos\alpha \) and \( \sin\beta \) only.

Cosine

The cosine of an angle in a right-angled triangle is defined as the adjacent side over the hypotenuse. In a broader sense, and particularly in the context of this problem, the cosine function relates the angle to the ratio of the length of adjacent side to the hypotenuse in a unit circle.

An important aspect to remember about the cosine function, especially in this exercise, is that it can take on positive and negative values depending on the quadrant in which the angle lies. In the fourth quadrant, where \( \alpha \) lies, the cosine of an angle is positive, but since we're looking for \( \sin\alpha \) to use in our formula, we must consider that the sine function is negative in the fourth quadrant. This knowledge is crucial when determining the exact values required for our problem.

Sine

The sine of an angle is defined, in the context of a right-angled triangle, as the ratio of the opposite side to the hypotenuse. Just like the cosine, the sine function can also be visualized in the unit circle, where it represents the y-coordinate of the point where the terminal side of the angle intersects the circle.

When working with the sine function, it's important to consider the angle's quadrant to determine the sign of the sine value. In the problem we are dealing with, \( \sin\beta \) is negative because \( \beta \) is in the third quadrant where both sine and cosine are negative. This gives us a starting point for computing the sine of the sum of the two angles using the sum of angles formula.

Tangent

Tangent is another trigonometric function that can be thought of as the ratio of \( \sin \theta \) to \( \cos \theta \) for a given angle \( \theta \) in a right-angled triangle.

• The formula for tangent of a sum of two angles is derived from the sine and cosine sum formulas and is given by \( \tan(\alpha + \beta) = \frac{\sin(\alpha + \beta)}{\cos(\alpha + \beta)}\). This is particularly useful when direct calculation of tangent is complex.

In our exercise, once we have determined \( \sin(\alpha + \beta) \) and \( \cos(\alpha + \beta) \) using their respective sum formulas, we can find \( \tan(\alpha + \beta) \) by dividing the sine by the cosine. Here, we must also bear in mind the signs of sine and cosine to ensure the correct sign of the tangent, which in this case will be negative as the sine and cosine of \( \alpha + \beta \) have different signs.