Problem 3 - Contemporary Precalculus

Question

Find \(\sin 2 x, \cos 2 x,\) and \(\tan 2 x\) under the given conditions. $$\cos x=-\frac{3}{5} \quad\left(\pi

Step-by-Step Solution

Answer

Given that \(\cos{x} = -\frac{3}{5}\) and the interval \(x\) lies in is \(\pi

Step 1: Determine the sign

Given that \(x\) lies in the interval \(\pi < x < \frac{3\pi}{2}\), we know that both \(\sin x\) and \(\cos x\) must be negative. Therefore, the value of \(\sin x\) is also negative.

Step 2: Use the Pythagorean identity

We will use the Pythagorean identity \(\sin^2 x + \cos^2 x = 1\) to find the value of \(\sin x\). Given that \(\cos x = -\frac{3}{5}\), we get: $$\sin^2 x + \left(-\frac{3}{5}\right)^2 = 1$$ $$\sin^2 x = 1 - \frac{9}{25}$$ $$\sin^2 x = \frac{16}{25}$$

Step 3: Take the square root

Since \(\sin x\) is negative, when we take the square root of both sides, we get: $$\sin x = -\frac{4}{5}$$ Step 2: Calculate the expressions

Step 4: \(\sin 2x\)

Using the double-angle formula, we have: $$\sin{2x} = 2\sin{x}\cos{x} = 2\left(-\frac{4}{5}\right)\left(-\frac{3}{5}\right) = \frac{24}{25}$$

Step 5: \(\cos 2x\)

Using the double-angle formula, we have: $$\cos{2x} = \cos^2{x}-\sin^2{x} = \left(-\frac{3}{5}\right)^2 - \left(-\frac{4}{5}\right)^2 = -\frac{7}{25}$$

Step 6: \(\tan x\)

Using the quotient identities \(\tan x = \frac{\sin x}{\cos x}\), we have: $$\tan{x} = \frac{-\frac{4}{5}}{-\frac{3}{5}} = \frac{4}{3}$$

Step 7: \(\tan 2x\)

Using the double-angle formula, we have: $$\tan{2x} = \frac{2\tan{x}}{1-\tan^2{x}} = \frac{2\left(\frac{4}{3}\right)}{1-\left(\frac{4}{3}\right)^2} = \frac{24}{7}$$ In conclusion, the values of the expressions are: $$\sin{2x} = \frac{24}{25}$$ $$\cos{2x} = -\frac{7}{25}$$ $$\tan{2x} = \frac{24}{7}$$

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is essential for simplifying and solving problems involving trigonometric functions. These identities are equations that relate the trigonometric functions to one another, allowing us to manipulate and transform complex expressions into simpler forms. One of the most fundamental sets of these identities are the double-angle formulas. They express the sine, cosine, and tangent of double angles (e.g., \(2x\)) in terms of the trigonometric functions of the angle itself (e.g., \(x\)).

For instance, the double-angle formula for sine is \(\sin(2x) = 2\sin(x)\cos(x)\), while for cosine, it is \(\cos(2x) = \cos^2(x) - \sin^2(x)\) or alternatively \(\cos(2x) = 2\cos^2(x) - 1\) or \(\cos(2x) = 1 - 2\sin^2(x)\). For the tangent, the double-angle formula is \(\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}\). These double-angle formulas become powerful tools when we need to find the trigonometric values of double angles, which appeared in the original exercise.

Pythagorean Identity

The Pythagorean identity is a cornerstone of trigonometry and is derived from the Pythagorean theorem. It states that for any angle \(x\), \(\sin^2(x) + \cos^2(x) = 1\). In our given problem, this identity allowed us to calculate the value of \(\sin x\) from the known value of \(\cos x\).

By substituting \(\cos x = -\frac{3}{5}\), we used the identity to find \(\sin x\). Since \(\sin^2 x + \left(-\frac{3}{5}\right)^2 = 1\), we found that \(\sin x = -\frac{4}{5}\) after taking into account the sign of \(\sin x\) in the specific quadrant. The Pythagorean identity is a versatile tool, especially when one trigonometric value is known, and we need to find another.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent are fundamental in understanding relationships within right-angled triangles and circles. In the context of the given exercise, knowing the signs of these functions in different quadrants is crucial. The angle \(x\) in the exercise falls in the third quadrant, where both sine and cosine are negative. This insight was key for correctly determining the value of \(\sin x\) and thus influences the sign when computing \(\sin 2x\), \(\cos 2x\), and \(\tan 2x\) using their respective double-angle formulas.

Recognizing the quadrant where the angle resides helps in understanding the behavior of trigonometric functions which then informs the sign of the final answer. It's also important to be familiar with the unit circle, as it provides a visual aid in determining the signs and values of the trigonometric functions at various angles.

Quotient Identities

Quotient identities are another class of trigonometric identities that relate sine and cosine to tangent and cotangent. The most commonly used quotient identity is \(\tan(x) = \frac{\sin(x)}{\cos(x)}\), which we applied in step 6 to find \(\tan(x)\) from the previously calculated values of \(\sin(x)\) and \(\cos(x)\).

In the exercise, by substituting the determined values of \(\sin x = -\frac{4}{5}\) and \(\cos x = -\frac{3}{5}\), we were able to compute \(\tan x = \frac{4}{3}\). This value played a crucial role in finding \(\tan 2x\) using the double-angle formula for tangent. The quotient identities are particularly useful when we need to find the tangent or cotangent function and we only have information about sine and cosine and vice versa.

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Problem 2

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Problem 3

Other exercises in this chapter

Problem 2

Find all solutions of the equation. $$\sin x=.682$$

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Problem 2

$$\text {Find the exact value.}$$ $$\tan \frac{\pi}{12}$$

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Problem 3

Find the exact functional value without using a calculator: $$\tan ^{-1}(-1)$$

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Problem 3

Find all solutions of the equation. $$\cos x=-.564$$

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Key Concepts

Trigonometric Identities

Understanding trigonometric identities is essential for simplifying and solving problems involving trigonometric functions. These identities are equations that relate the trigonometric functions to one another, allowing us to manipulate and transform complex expressions into simpler forms. One of the most fundamental sets of these identities are the double-angle formulas. They express the sine, cosine, and tangent of double angles (e.g., \(2x\)) in terms of the trigonometric functions of the angle itself (e.g., \(x\)).

For instance, the double-angle formula for sine is \(\sin(2x) = 2\sin(x)\cos(x)\), while for cosine, it is \(\cos(2x) = \cos^2(x) - \sin^2(x)\) or alternatively \(\cos(2x) = 2\cos^2(x) - 1\) or \(\cos(2x) = 1 - 2\sin^2(x)\). For the tangent, the double-angle formula is \(\tan(2x) = \frac{2\tan(x)}{1 - \tan^2(x)}\). These double-angle formulas become powerful tools when we need to find the trigonometric values of double angles, which appeared in the original exercise.

Pythagorean Identity

The Pythagorean identity is a cornerstone of trigonometry and is derived from the Pythagorean theorem. It states that for any angle \(x\), \(\sin^2(x) + \cos^2(x) = 1\). In our given problem, this identity allowed us to calculate the value of \(\sin x\) from the known value of \(\cos x\).

By substituting \(\cos x = -\frac{3}{5}\), we used the identity to find \(\sin x\). Since \(\sin^2 x + \left(-\frac{3}{5}\right)^2 = 1\), we found that \(\sin x = -\frac{4}{5}\) after taking into account the sign of \(\sin x\) in the specific quadrant. The Pythagorean identity is a versatile tool, especially when one trigonometric value is known, and we need to find another.

Trigonometric Functions

Trigonometric functions such as sine, cosine, and tangent are fundamental in understanding relationships within right-angled triangles and circles. In the context of the given exercise, knowing the signs of these functions in different quadrants is crucial. The angle \(x\) in the exercise falls in the third quadrant, where both sine and cosine are negative. This insight was key for correctly determining the value of \(\sin x\) and thus influences the sign when computing \(\sin 2x\), \(\cos 2x\), and \(\tan 2x\) using their respective double-angle formulas.

Recognizing the quadrant where the angle resides helps in understanding the behavior of trigonometric functions which then informs the sign of the final answer. It's also important to be familiar with the unit circle, as it provides a visual aid in determining the signs and values of the trigonometric functions at various angles.

Quotient Identities

Quotient identities are another class of trigonometric identities that relate sine and cosine to tangent and cotangent. The most commonly used quotient identity is \(\tan(x) = \frac{\sin(x)}{\cos(x)}\), which we applied in step 6 to find \(\tan(x)\) from the previously calculated values of \(\sin(x)\) and \(\cos(x)\).

In the exercise, by substituting the determined values of \(\sin x = -\frac{4}{5}\) and \(\cos x = -\frac{3}{5}\), we were able to compute \(\tan x = \frac{4}{3}\). This value played a crucial role in finding \(\tan 2x\) using the double-angle formula for tangent. The quotient identities are particularly useful when we need to find the tangent or cotangent function and we only have information about sine and cosine and vice versa.