Problem 4 - Calculus: An Integrated Approach to Functions and Their Rates of Change

Question

(a) Find an acute angle \(x\) such that \(\sin x=0.5\). (b) Find all \(x\) between 0 and \(2 \pi\) such that \(\sin x=0.5\). (c) Find an acute angle \(x\) such that \(\cos x=0.5\). (d) Find all \(x \in[0,2 \pi]\) such that \(\cos (2 x)=0.5\).

Step-by-Step Solution

Answer

The answers are: (a) \(x = \pi / 6\), (b) \(x = \pi/6, 5\pi/6\), (c) \(x = \pi / 3\), (d) \(x = \pi/6, 5\pi/6\).

Step 1: Solve for \(x\) when \(\sin x = 0.5\)

First, you need to find the angle whose sine is 0.5. One such angle is \(x = \pi / 6\) or \(30^\circ\). This is because sine of \(30^\circ\) or \(\pi/6\) is known to be 0.5.

Step 2: Find all \(x\) between 0 and \(2\pi\) where \(\sin x = 0.5\)

Since \(\sin x = \sin(x + 2n\pi)\), for all integers \(n\), and \(\sin x = \sin(\pi - x)\), the solutions in the interval [0 , \(2\pi\)] are \(x = \pi/6, 5\pi/6\)

Step 3: Solve for \(x\) when \(\cos x = 0.5\)

Next, you need to consider the angle whose cosine is 0.5. One such acute angle is \(x = \pi / 3\) or \(60^\circ\). This is because cosine of \(60^\circ\) or \(\pi/3\) is known to be 0.5.

Step 4: Find all \(x\) between 0 and \(2\pi\) where \(\cos(2x) = 0.5\)

This problem can be translated to finding \(x\) such that \(\cos y = 0.5\), where \(y = 2x\). From step 3, we know that \(y=\frac{\pi}{3}, \frac{5\pi}{3}\), so \(x=\frac{\pi}{6}, \frac{5\pi}{6}\). But because the periodicity of the cosine function is \(2\pi\), and in the given range there could be more solutions, we need to add or subtract multiples of \(\pi\) to/from these. But even after adding or subtracting any multiple of \(\pi\), all values will still be either \(\frac{\pi}{6}\) or \(\frac{5\pi}{6}\)

Key Concepts

Sine Function

The sine function is one of the fundamental trigonometric functions used in mathematics. It is defined for an angle \( x \) as the ratio of the length of the opposite side to the length of the hypotenuse in a right-angled triangle.

• For acute angles, the sine function will always produce values between 0 and 1.
• The function is periodic, with a period of \( 2\pi \). This means it repeats its values every \( 2\pi \) radians.

In the exercise, finding an angle \( x \) such that \( \sin x = 0.5 \) involves knowing common angle values. A key angle where this is true is \( x = \pi/6 \) or 30°.

Cosine Function

Much like sine, the cosine function is another critical trigonometric function. It is defined for an angle \( x \) as the ratio of the adjacent side to the hypotenuse in a right-angled triangle. Here are some key points:

• The cosine function creates values that are between -1 and 1 for any real angle \( x \).
• It is also periodic with a period the same as sine, which is \( 2\pi \).

In the context of the exercise, to find an acute angle \( x \) such that \( \cos x = 0.5 \), the angle can be \( \pi/3 \) or 60°.

Angle Solutions

When solving trigonometric equations like \( \sin x = 0.5 \) or \( \cos x = 0.5 \), we need to find all possible angles \( x \) that fit the condition. This is called finding angle solutions.For instance, when the sine of \( x \) is 0.5:

• Using the property \( \sin(\pi - x) = \sin x \), we find the additional solution \( x = 5\pi/6 \) within a \( 0 \to 2\pi \) interval.

Similar properties and transformations apply to the cosine function to find comprehensive solutions.

Trigonometric Equations

Trigonometric equations are equations involving trigonometric functions like sine, cosine, or tangent. To solve these equations, we typically apply identities and properties of the trigonometric functions:

• Use identities like \( \cos(2x) = \cos x \) to simplify and solve the equation.
• Identify and use known function values for standard angles in radians to find solutions.

For \( \cos(2x) = 0.5 \) in the exercise, solving requires understanding the periodic nature of the cosine. This involves finding initial solutions, then addressing the possible addition of multiples of \( \pi \) to find all solutions within a given range.