Problem 131 - 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) \(g(-\boldsymbol{\theta})\) $$\theta=-270^{\circ}$$

Step-by-Step Solution

Answer

(a) 1, (b) -1, (c) 0, (d) 0, (e) 0, (f) 0.

Step 1: Calculate (f+g)(θ)

Substitute θ with -270° in f(θ)=sinθ and g(θ)=cosθ. \n f(-270°) = sin(-270°) = 1 (since the sine of -270° is 1), and g(-270°) = cos(-270°) = 0 (since the cosine of -270° is 0). Then calculate (f+g)(-270°) which is 1 + 0 = 1.

Step 2: Calculate (g-f)(θ)

We use the values calculated in the previous step. Then, the difference (g - f) (-270°) becomes 0 - 1 = -1.

Step 3: Calculate [g(θ)]^2

Square g(-270°) which is 0. So, [g(-270°)]^2 = 0^2 = 0.

Step 4: Calculate (fg)(θ)

Multiply the results of f(θ) and g(θ). So, (fg)(-270°) becomes 1 * 0 = 0.

Step 5: Calculate f(2θ)

Substitute 2θ with 2(-270°)=-540° in f(θ). Remembering the period of sine is 360°, we simplify −540° to −540° + 360°*2 = 180°. Therefore, f(-540°)=sin(180°)=0.

Step 6: Calculate g(-θ)

Substitute −θ with -(-270°)=270° into g(θ). Remember that cosine is an even function which means cos(θ) = cos(−θ). Hence, g(-(-270°))=cos(270°)=0.

Key Concepts

Sine Function

The sine function, represented by \( \sin \theta \), is a fundamental trigonometric function that measures the y-coordinate of a point on the unit circle as it relates to a given angle. The sine function repeats every 360 degrees, which means it has a period of 360°.

This cyclic nature of sine can be observed in its wave-like graph, oscillating between -1 and 1. It's important to remember:

• At 0° and 180°, \( \sin \theta \) is 0.
• It reaches its maximum value of 1 at 90° and minimum of -1 at 270°.
• The sine of negative angles can be calculated using \( \sin(-\theta) = -\sin(\theta) \).

If you are trying to calculate \( \sin(-270°) \), keep in mind these principles, especially that the sine of -270° equals 1.

Cosine Function

The cosine function, denoted as \( \cos \theta \), is another key trigonometric function. It corresponds to the x-coordinate of a point on the unit circle. The cycle of cosine, just like sine, repeats every 360 degrees. Cosine is distinguished by:

• Being 1 at 0°, and -1 at 180°.
• Having zeros at 90° and 270°.
• Being an even function, \( \cos(\theta) = \cos(-\theta) \).

In the problem, for \( \cos(-270°) \), it equates to 0. This is due to the symmetrical nature of the cosine function along the y-axis.

Exact Values

Exact values refer to specific outcomes of trigonometric functions without using a calculator, typically at standard angles such as 0°, 30°, 45°, 60°, 90°, and their multiples. These can be derived from the unit circle and fundamental identities such as Pythagorean identities.

Knowing these values can help compute expressions swiftly and is often tackled by memorizing or understanding triangles and their angles. For example:

• \( \sin(90°) = 1 \), \( \cos(90°) = 0 \).
• \( \sin(180°) = 0 \), \( \cos(180°) = -1 \).
• Symmetry and periodicity help in deriving values for negative angles or ones beyond 360°.

In problems like the one given, compute exact values such as \( \sin(-270°) \) and \( \cos(-270°) \) by leveraging these known values and patterns.

Angle Conversion

Angle conversion is essential for dealing with trigonometric functions, particularly when angles are not in standard form. The process involves changing angles into measures easier to evaluate, such as converting a negative angle or one larger than 360°.

This can be done by:

• Adding or subtracting multiples of 360° to bring angles within the 0° to 360° range.
• Using known angle identities to simplify calculations.
• Knowing that, for sine and cosine, the repeating pattern helps to compute angles larger than 360° efficiently.

For example, converting \( -540° \) into an equivalent angle, you would add 720° (2*360°), resulting in \( 180° \). Understanding this helps when calculating functions like \( \sin(-540°) \) by simplifying them into recognizable angles.