Problem 124 - 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) \(g(-\boldsymbol{\theta})\) $$\theta=225^{\circ}$$

Step-by-Step Solution

Answer

(a) The result is -\sqrt{2}. (b) The result is 0. (c) The result is 1/2. (d) The result is 1/2. (e) The result is 1. (f) The result is -\sqrt{2}/2.

Step 1: Solve Part (a)

For part (a), the required operation is \(f(\theta) + g(\theta) = \sin \theta + \cos \theta\). Just substitute \(\theta = 225^{\circ}\) in this equation. We know from the unit circle that \(\sin 225^{\circ} = -\sqrt{2}/2\) and \(\cos 225^{\circ} = -\sqrt{2}/2\). So, \((f+g)(\theta) = \sin 225^{\circ} + \cos 225^{\circ} = -\sqrt{2}/2 - \sqrt{2}/2 = -\sqrt{2}\).

Step 2: Solve Part (b)

For part (b), the required operation is \(g(\theta) - f(\theta) = \cos \theta - \sin \theta\). Substitute \(\theta = 225^{\circ}\) in this equation. Using the values from the unit circle, we get \((g-f)(\theta) = \cos 225^{\circ} - \sin 225^{\circ} = -\sqrt{2}/2 - (-\sqrt{2}/2) = 0\).

Step 3: Solve Part (c)

For part (c), the required operation is \([g(\theta)]^{2} = [\cos \theta]^{2}\). Substitute \(\theta = 225^{\circ}\) in this equation. Using the values from the unit circle we obtain \([g(\theta)]^{2} = [\cos 225^{\circ}]^{2} = [ -\sqrt{2}/2]^{2} = 1/2\).

Step 4: Solve Part (d)

For part (d), the required operation is \(f(\theta) * g(\theta) = \sin \theta * \cos \theta\). Substitute \(\theta = 225^{\circ}\) in this equation. Using the values from the unit circle we get \((f g)(\theta) = \sin 225^{\circ} * \cos 225^{\circ} = (-\sqrt{2}/2) * (-\sqrt{2}/2) = 1/2\).

Step 5: Solve Part (e)

For part (e), the required operation is \(f(2\theta) = \sin 2\theta\). Substitute \(\theta = 225^{\circ}\) in this equation, which results in solving for \(\sin 450^{\circ}\). But, since sin function has a period of \(360^{\circ}\), \(\sin 450^{\circ} = \sin 90^{\circ} = 1\). So, f(2\(\theta)) = 1.

Step 6: Solve Part (f)

For part (f), the required operation is \(g(-\theta) = \cos -\theta\). Substitute \(\theta = 225^{\circ}\) in this equation, which results in solving for \(\cos -225^{\circ}\). The cos function is even, hence \(\cos -\theta = \cos \theta\), so \(\cos -225^{\circ} = \cos 225^{\circ} = -\sqrt{2}/2\). So, g(-\(\theta)) = -\sqrt{2}/2.

Key Concepts

Unit Circle

The unit circle is a powerful tool in trigonometry used to understand the behavior of trigonometric functions at various angles. Consider the unit circle as a circle with a radius of 1, centered at the origin of a coordinate plane.
Each point on the circle corresponds to a specific angle, measured in degrees or radians, from a central point known as the origin. This allows us to establish relationships between angles and their sine and cosine values.
For example, the angle 225° falls in the third quadrant of the unit circle, where both sine and cosine values are negative. These values are derived from the coordinates of the point where the angle’s terminal side intersects the circle.

• In the unit circle, the x-coordinate represents the cosine value of the angle.
• The y-coordinate represents the sine value of the angle.

This makes the unit circle a crucial reference for determining exact values of trigonometric functions without a calculator.

Sine Function

The sine function is a fundamental trigonometric function that gives us the y-coordinate of a point on the unit circle for any given angle. It is commonly expressed as \( sin \, \theta \).
As angles rotate around the unit circle, the value of the sine function changes, ranging from -1 to 1.
Key properties of the sine function include:

• Sine is positive in the first and second quadrants.
• Sine is negative in the third and fourth quadrants.
• It has a period of 360°, meaning it repeats its pattern every 360°.

For the angle 225°, the sine function yields -\sqrt{2}/2. This is because at 225°, you are in the third quadrant where sine values are negative, and the unit circle provides these concrete values.

Cosine Function

The cosine function, often expressed as \(\cos \theta\), provides the x-coordinate of a point on the unit circle. Like the sine function, cosine values range between -1 and 1.
It also exhibits specific behavior depending on the quadrant of the angle:

• Cosine is positive in the first and fourth quadrants.
• Cosine is negative in the second and third quadrants.
• It also has a periodicity of 360°.

For the angle 225°, cosine equals -\sqrt{2}/2. At this angle, you are in the third quadrant where cosine values become negative, similar to sine.

Angle Operations

Trigonometric angle operations involve using trigonometric identities and properties to determine the function's values for different alterations of a given angle, without needing a calculator.
Common angle operations include:

• Addition and subtraction of angles, so you can find \(sin(\theta + \phi)\) or \(cos(\theta - \phi)\).
• Multiplication of angles for double-angle formulas, such as \(sin(2\theta)\).
• Using symmetry properties, such as the even property of the cosine function, which states \(cos(-\theta) = cos(\theta)\).

For example, at \(\theta = 225°\), simple operations like \(\sin(2\theta)\) are solved using double-angle formulas. Here, \(2\theta\) equals 450°, and remembering the periodic nature, \(\sin(450°) = \sin(90°) = 1\). Negative angles for cosine are handled using its even property, resulting in the same value as their positive counterparts.