Problem 122 - 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) \(g(-\boldsymbol{\theta})\) $$\theta=60^{\circ}$$

Step-by-Step Solution

Answer

(a) \((f+g)(60°) = \sqrt{3}/2 + 1/2\), (b) \((g-f)(60°) = 1/2 - \sqrt{3}/2\), (c) \([g(60°)]² = (1/2)²\), (d) \((fg)(60°) = (\sqrt{3}/2) \cdot (1/2)\), (e) \(f(2 \cdot 60°) = \sqrt{3}/2\), (f) \(g(-60°) = 1/2\).

Step 1: (a) (f+g)(θ)

First, we need to find the sum of \(f(\theta) = \sin\theta\) and \(g(\theta) = \cos\theta\). At \(θ = 60°\), we know from basic trigonometry that, \(\sin(60°) = \sqrt{3}/2\) and \(\cos(60°) = 1/2\). So, \((f+g)(60°) = \sin(60°) + \cos(60°) = \sqrt{3}/2 + 1/2\).

Step 2: (b) (g-f)(θ)

Now, we need to find the difference between \(g(\theta) = \cos\theta\) and \(f(\theta) = \sin\theta\). So, \((g-f)(60°) = \cos(60°) - \sin(60°) = 1/2 - \sqrt{3}/2\).

Step 3: (c) [g(θ)]²

Next, we need to find the square of \(g(\theta) = \cos\theta\). So, \([g(60°)]² = [\cos(60°)]² = (1/2)²\).

Step 4: (d) (fg)(θ)

The next task is to find the product of \(f(\theta) = \sin\theta\) and \(g(\theta) = \cos\theta\). So, \((fg)(60°) = \sin(60°) \cdot \cos(60°) = (\sqrt{3}/2) \cdot (1/2)\).

Step 5: (e) f(2θ)

Now, we need to find \(f(2θ) = \sin(2θ)\). But there is a trigonometric identity that might help here: \(\sin(2θ) = 2 \sin(θ) \cos(θ)\). So, \(f(2 \cdot 60°) = \sin(120°) = \sqrt{3}/2\).

Step 6: (f) g(-θ)

Finally, we need to find \(g(-θ) = \cos(-θ)\). Because cosine is an even function, it possesses the property \(\cos(-θ) = \cos(θ)\). So, \(g(-60°) = \cos(-60°) = \cos(60°) = 1/2\).

Key Concepts

Sine Function

The Sine Function, often denoted as \( \sin \theta \), is a fundamental trigonometric function. It relates the angle \( \theta \) in a right triangle to the ratio of the length of the opposite side to the hypotenuse. This function is periodic, with a period of \( 360° \) or \( 2\pi \) radians, meaning it repeats its values in these intervals. The sine of \( 60° \) is particularly important and is often remembered by heart:

• \( \sin(60°) = \sqrt{3}/2 \)

This value is derived from the properties of an equilateral triangle, where each angle is \( 60° \), and the height divides it into two 30-60-90 right triangles. Understanding this function is crucial for solving trigonometric problems involving angles and can provide insights into wave patterns in physics.

Cosine Function

The Cosine Function, denoted as \( \cos \theta \), is another key trigonometric function, complementing the sine function. It relates the angle \( \theta \) to the ratio of the adjacent side to the hypotenuse in a right triangle. Like the sine function, cosine is periodic with the same interval of repetition. This makes it a fundamental concept in studying periodic motions. For the angle of \( 60° \), the cosine function takes on a specific and regularly cited value:

• \( \cos(60°) = 1/2 \)

This value emerges from examining the same 30-60-90 triangle formed in an equilateral triangle. Mastering the cosine function is essential for understanding the behavior of waves and oscillatory systems.

Trigonometric Identities

Trigonometric Identities are mathematical statements that relate different trigonometric functions to one another. These identities are used to simplify expressions and solve equations involving trigonometric functions. A crucial identity that arises in the exercise is the double-angle identity:

• \( \sin(2\theta) = 2\sin(\theta)\cos(\theta) \)

This particular identity helps in finding the sine of double an angle, such as \( \sin(120°) \), without direct measurement. In the exercise, using this identity simplifies the calculation of \( \sin(120°) \). Trigonometric identities like this one are vital tools for students in mathematics, physics, and engineering.

Trigonometric Values

Trigonometric Values are specific numerical outcomes derived from trigonometric functions at predefined angles. These values can often be derived from specific geometric constructions or can be memorized for rapid use. For common angles such as \( 30° \), \( 45° \), \( 60° \), and \( 90° \), students should know:

• \( \sin(30°) = 1/2 \)
• \( \sin(45°) = \sqrt{2}/2 \)
• \( \sin(60°) = \sqrt{3}/2 \)
• \( \cos(30°) = \sqrt{3}/2 \)
• \( \cos(45°) = \sqrt{2}/2 \)
• \( \cos(60°) = 1/2 \)

Understanding these typical values is crucial for accurately computing expressions without a calculator, as seen in many math and physics problems.

Angle Measures

Angle Measures can be expressed in degrees or radians, with a full circle equating to \( 360° \) or \( 2\pi \) radians. Knowing how to convert between these two systems is vital for trigonometry. For example, \( 60° \) can be expressed as \( \pi/3 \) radians. In trigonometry, problems frequently require recognition of both systems. Keeping these conversions in mind is essential for applying trigonometric functions correctly in various contexts, whether working in theoretical math or practical applications in physics or engineering. Familiarity with these measures allows seamless navigation through problems requiring the calculation or manipulation of angles.