Problem 27 - Contemporary Precalculus

Question

Simplify the given expression. $$\text { If } \cos x=-\frac{1}{5} \text { and } \pi

Step-by-Step Solution

Answer

Answer: The value of sin(π/3 - x) is $\frac{2\sqrt{6}-\sqrt{3}}{10}$.

Step 1: Recall the sine subtraction formula.

The sine subtraction formula is: $$\sin(A - B) = \sin A \cos B - \cos A \sin B$$ In this problem, we have A = π/3 and B = x, so we want to find the value of sin(π/3 - x).

Step 2: Find the value of sin(x) using the Pythagorean identity.

The Pythagorean identity states that: $$\sin^2x + \cos^2x = 1$$ We are given the value of cos(x), which is -1/5. Now we can find the value of sin(x): $$\sin^2x + (-\frac{1}{5})^2 = 1$$ $$\sin^2x = 1 - \frac{1}{25}$$ $$\sin^2x = \frac{24}{25}$$ Since x is in the third quadrant (π < x < 3π/2), sin(x) is negative. So, $$\sin x = -\sqrt{\frac{24}{25}}$$

Step 3: Find the values of sin(π/3) and cos(π/3).

We need these values to use the sine subtraction formula. $$\sin{\frac{\pi}{3}} = \frac{\sqrt{3}}{2}$$ $$\cos{\frac{\pi}{3}} = \frac{1}{2}$$

Step 4: Apply the sine subtraction formula.

Now we can find the value of sin(π/3 - x) using the sine subtraction formula: $$\sin{\left(\frac{\pi}{3} - x\right)} = \sin{\frac{\pi}{3}} \cos x - \cos{\frac{\pi}{3}} \sin x$$ Inputting the known values for sin(π/3), cos(π/3), sin(x), and cos(x), we get: $$\sin{\left(\frac{\pi}{3} - x\right)} = \frac{\sqrt{3}}{2}\cdot\left(-\frac{1}{5}\right) - \frac{1}{2}\cdot\left(-\sqrt{\frac{24}{25}}\right)$$

Step 5: Simplify the expression.

Calculate the expression: $$\sin{\left(\frac{\pi}{3} - x\right)} = -\frac{\sqrt{3}}{10} + \frac{1}{2}\sqrt{\frac{24}{25}}$$ $$\sin{\left(\frac{\pi}{3} - x\right)} = -\frac{\sqrt{3}}{10} + \frac{2\sqrt{6}}{10}$$ Now, combine the terms to get the final answer: $$\sin{\left(\frac{\pi}{3} - x\right)} = \frac{2\sqrt{6}-\sqrt{3}}{10}$$

Key Concepts

Sine Subtraction Formula

To solve expressions such as \( \sin(\frac{\pi}{3} - x) \), we need to understand the Sine Subtraction Formula. This formula is an extension of trigonometric identities and helps in finding the sine of an angle formed by the subtraction of two known angles. The formula is given by:

• \( \sin(A - B) = \sin A \cos B - \cos A \sin B \)

When applying this formula to specific angles, substituting \( A = \frac{\pi}{3} \) and \( B = x \) lets us express \( \sin(\frac{\pi}{3} - x) \) in terms of \( \sin \frac{\pi}{3}, \cos \frac{\pi}{3}, \sin x, \text{ and } \cos x \). This approach is especially useful because it transforms a complex expression into manageable algebraic components. Breaking problems into these known values allows for precise calculations even when direct measurement of the angle isn’t possible.

Pythagorean Identity

A fundamental concept in trigonometry is the Pythagorean Identity. It expresses an intrinsic relationship between sine and cosine for any angle \( x \). The identity is mathematically expressed as:

• \( \sin^2 x + \cos^2 x = 1 \)

This allows us to find the sine of an angle when its cosine is known, and vice versa. For instance, if \( \cos x = -\frac{1}{5} \), we can rearrange the identity to find \( \sin^2 x = 1 - \cos^2 x \). Further solving gives \( \sin^2 x = \frac{24}{25} \). The quadrants dictate the sign of \( \sin x \), and in this case, since \( x \) is in the third quadrant where sine is negative, we find \( \sin x = -\sqrt{\frac{24}{25}} \). Understanding the Pythagorean Identity is crucial in simplifying trigonometric expressions, especially when dealing with complex angle measures or in contexts where only one trigonometric value is known.

Angle Quadrants

In trigonometry, the coordinate plane is divided into four quadrants that are important for determining the sign of sine, cosine, and tangent functions.

• The first quadrant \((0, \frac{\pi}{2})\) is where all trigonometric functions are positive.
• In the second quadrant \((\frac{\pi}{2}, \pi)\), sine is positive while cosine and tangent are negative.
• In the third quadrant \((\pi, \frac{3\pi}{2})\), sine and cosine are both negative, while tangent is positive.
• Finally, in the fourth quadrant \((\frac{3\pi}{2}, 2\pi)\), cosine is positive and sine and tangent are negative.

For the given problem, since \( x \) is in the third quadrant, we know that \( \cos x = -\frac{1}{5} \) and \( \sin x = -\sqrt{\frac{24}{25}} \). This understanding of quadrants provides a systematic method to determine the correct sign when evaluating trigonometric identities, ensuring the calculations align with the angle's actual position on the plane.