Problem 125 - Algebra and Trigonometry Real Mathematics, Real People

Question

Find the exact value of each function for the given angle for \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta .\) Do not use a calculator. (a) \((f+g)(\theta)\) (b) \((g-f)(\theta)\) (c) \([g(\theta)]^{2}\) (d) \((f g)(\theta)\) (e) \(f(2 \theta)\) (f) \(g(-\boldsymbol{\theta})\) $$\theta=-150^{\circ}$$

Step-by-Step Solution

Answer

(a) (f+g)(θ) = -\(\frac{1+\sqrt{3}}{2} \) , (b) (g-f)(θ) = -\(\frac{\sqrt{3}-1}{2}\) , (c) [g(θ)]²= \(\frac{3}{4}\) , (d) (fg)(θ)= \(\frac{\sqrt{3}}{4}\), (e) f(2θ) cannot be calculated with the provided data, (f) g(-θ)= \(-\frac{\sqrt{3}}{2}\)

Step 1: Determine sine and cosine values for the angle

The provided angle is -150 degrees, which is equivalent to 210 degrees if we go around the unit circle in the positive direction, because a full circle is 360 degrees. Therefore, \(\sin(-150^{\circ})=\sin(210^{\circ})=-\frac{1}{2}\) and \(\cos(-150^{\circ})=\cos(210^{\circ})=-\frac{\sqrt{3}}{2}\).

Step 2: Compute (f+g)(θ)

Plug the values of \(f(\theta)=\sin \theta\) and \(g(\theta)=\cos \theta\) into the equation for (f+g)(θ). Therefore, (f+g)(θ)= \(-\frac{1}{2}-\frac{\sqrt{3}}{2}=-\frac{1+\sqrt{3}}{2} \).

Step 3: Compute (g-f)(θ)

Using the values found for \(f(\theta)\) and \(g(\theta)\), compute the expression for (g-f)(θ). Therefore, (g-f)(θ)= \(-\frac{\sqrt{3}}{2}-(-\frac{1}{2})=-\frac{\sqrt{3}-1}{2} \).

Step 4: Compute [g(θ)]²

Compute the square of \(g(\theta)=\cos \theta\). Therefore, [g(θ)]²= \(\left(-\frac{\sqrt{3}}{2}\right)^2=\frac{3}{4}\).

Step 5: Compute (fg)(θ)

Multiply the values of \(f(\theta)\) and \(g(\theta)\). Therefore, (fg)(θ)= \(-\frac{1}{2} \cdot -\frac{\sqrt{3}}{2}=\frac{\sqrt{3}}{4}\).

Step 6: Compute f(2θ)

Calculate the sine of double the angle. However, \(\sin(2\theta)=2\sin\theta\cos\theta\) and we cannot calculate it without knowing the exact value of \(\sin(420^{\circ})\). Since this value is not provided, we can't compute the value.

Step 7: Compute g(-θ)

Calculate the cosine of the negative angle. We know that cosine is an even function, meaning \(\cos(-\theta)=\cos\theta\). Hence, g(-θ)= \(-\frac{\sqrt{3}}{2}\).

Key Concepts

Sine and Cosine Values

Understanding sine and cosine values is fundamental in trigonometry. These functions relate the angles of a right triangle to the lengths of its sides, specifically:

• The sine of an angle, represented as \( \sin \theta \) , is the ratio of the length of the opposite side to the hypotenuse.
• The cosine of an angle, represented as \( \cos \theta \), is the ratio of the length of the adjacent side to the hypotenuse.

In the unit circle, these values correspond to the y-coordinate and the x-coordinate, respectively, of a point positioned at an angle \( \theta \) from the positive x-axis. For example, at \( \theta = -150^{\circ} \), the unit circle equivalent is \( 210^{\circ} \) (moving positively), where \( \sin(210^{\circ}) = -\frac{1}{2} \) and \( \cos(210^{\circ}) = -\frac{\sqrt{3}}{2} \). These values help us understand the position of the terminal side of the angle and its relation to the x and y axes on the circle.

Angle Conversion

Angles in trigonometry can be measured in degrees or radians. Converting between these two units is crucial for solving various trigonometric problems.

• Degrees are more intuitive for everyday use, as a full circle is 360 degrees.
• Radians are often used in calculus as they provide a direct relationship between arc length and radius.

The conversion between the two is done using the formula: \[\theta_{\text{radians}} = \theta_{\text{degrees}} \times \frac{\pi}{180}\]For example, \( -150^{\circ} \) can be converted into radians by \( -150 \times \frac{\pi}{180} \), resulting in \( -\frac{5\pi}{6} \). Understanding how to convert angles is vital for applications involving calculus and complex trigonometric computations.

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for all angle values. They are powerful tools for simplifying and solving trigonometric problems.

• Pythagorean Identity: \( \sin^2\theta + \cos^2\theta = 1 \)
• Angle Sum and Difference Identities: Useful for computing exact values not readily available in standard tables.
• Double Angle Formulas: Such as \( \sin(2\theta) = 2\sin\theta\cos\theta \)

In the given problem, these identities can help check the computations or offer alternate methods to solve parts of the question. For instance, to compute \( f(2\theta) \), recognizing \( \sin(2\theta) = 2\sin\theta\cos\theta \) allows us to say: \[\sin(2 \times -150^{\circ}) = 2 \times \left(-\frac{1}{2}\right) \times \left(-\frac{\sqrt{3}}{2}\right) = \frac{\sqrt{3}}{2}\]

Negative Angles

Negative angles in trigonometry signify rotation in the clockwise direction on the unit circle. Understanding their behavior is crucial for comprehensively grasping the nuances of trigonometric functions.

• The sine function is odd, which means that \( \sin(-\theta) = -\sin(\theta) \)
• The cosine function is even, implying \( \cos(-\theta) = \cos(\theta)\)

For example, both \( \cos(-150^{\circ}) = \cos(150^{\circ}) \) and remain the same, which clarifies why we see certain symmetries in trigonometric values. Considering these properties can significantly simplify solving trigonometric problems involving negative angles, as even functions do not change the function's value. Additionally, understanding these symmetries aids in predicting trigonometric values for angles beyond the standard 0 to 360-degree range.