Problem 38 - Precalculus Mathematics for Calculus

Question

Find \(\sin \frac{x}{2}, \cos \frac{x}{2},\) and \(\tan \frac{x}{2}\) from the given information. $$\cos x=-\frac{4}{5}, \quad 180^{\circ}

Step-by-Step Solution

Answer

\(\sin \frac{x}{2} = \sqrt{\frac{9}{10}}, \cos \frac{x}{2} = -\sqrt{\frac{1}{10}}, \tan \frac{x}{2} = -3\).

Step 1: Determine the quadrant

The angle range \(180^{\circ}

Step 2: Use Pythagorean identity to find \(\sin x\)

Use the identity \(\sin^2 x + \cos^2 x = 1\). Since \(\cos x = -\frac{4}{5}\), substitute to get \(\sin^2 x = 1 - \left( -\frac{4}{5} \right)^2 = 1 - \frac{16}{25} = \frac{9}{25}\). Hence, \(\sin x = -\frac{3}{5}\) because sine is negative in the third quadrant.

Step 3: Use identities to find \(\sin \frac{x}{2}\) and \(\cos \frac{x}{2}\)

Use the half-angle identities: \(\sin \frac{x}{2} = \sqrt{\frac{1 - \cos x}{2}}\) and \(\cos \frac{x}{2} = -\sqrt{\frac{1 + \cos x}{2}}\) (negative because \(\frac{x}{2}\) is in the second quadrant where cosine is negative). Substitute \(-\frac{4}{5}\) for \(\cos x\) to get \(\sin \frac{x}{2} = \sqrt{\frac{1 + \frac{4}{5}}{2}} = \sqrt{\frac{9}{10}}\), and \(\cos \frac{x}{2} = -\sqrt{\frac{1 - \frac{4}{5}}{2}} = -\sqrt{\frac{1}{10}}\).

Step 4: Calculate \(\tan \frac{x}{2}\) using the identity

Use the identity \(\tan \frac{x}{2} = \frac{\sin \frac{x}{2}}{\cos \frac{x}{2}}\). Substitute the values obtained: \(\tan \frac{x}{2} = \frac{\sqrt{\frac{9}{10}}}{-\sqrt{\frac{1}{10}}} = -3\).

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in mathematics. They establish important relationships between trigonometric functions. One of the most fundamental trigonometric identities is the Pythagorean identity. This can be written as:

• \( \sin^2 x + \cos^2 x = 1 \)

This identity depicts how the squares of sine and cosine of an angle sum up to one. It forms the basis of many trigonometric calculations and problem-solving processes.

Another crucial set of identities involves half-angle identities. These identities help in finding the trigonometric functions of half of a given angle. For example:

• \( \sin \frac{x}{2} = \sqrt{\frac{1 - \cos x}{2}} \)
• \( \cos \frac{x}{2} = \sqrt{\frac{1 + \cos x}{2}} \)

These identities are especially useful when angles are not in conventional positions, like in the third or second quadrant. They require an understanding of the signs of the trigonometric functions in different quadrants.

Third Quadrant

The third quadrant of the Cartesian plane plays a vital role in trigonometry. It is defined by angles between \(180^\circ\) and \(270^\circ\). Since the angle \(x\) in the original problem falls within this range, it is situated in the third quadrant.

In this specific quadrant:

• Both sine and cosine are negative, which impacts the calculation of sine and cosine of any value of \(x\) in this range.

Knowing that cosine and sine are negative helps in evaluating trigonometric functions, especially when applied in half-angle identities.

By referring back to the quadrant details, one can confidently determine the signs of trigonometric outputs like \(\sin x = -\frac{3}{5} \) in the provided context.

Cosine Negative Angle

When dealing with trigonometric functions, understanding angles' nature like negative angles is imperative. Cosine of a negative angle, indicated as \(-\cos x\), refers to the direction of the cosine value when angles are defined on different quadrants.

In our problem, the cosine of \(x\) is \(-\frac{4}{5}\) as given because \(x\) is in the third quadrant. This reflects the inherent property that cosine functions flip their positivity or negativity according to the quadrant:

• In the first quadrant, cosine is positive.
• In the third quadrant (as in this problem), cosine values are negative.

When using cosine negative values in half-angle identities, it impacts calculations such as:

• \(\cos \frac{x}{2} = -\sqrt{\frac{1 + \cos x}{2}}\)

This ensures that the signs are correctly considered, leading to accurate evaluations.

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Problem 38

Other exercises in this chapter

Problem 37

Verify the identity. $$\tan \theta+\cot \theta=\sec \theta \csc \theta$$

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Problem 37

Prove the identity. $$\tan x-\tan y=\frac{\sin (x-y)}{\cos x \cos y}$$

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Problem 38

Find all solutions of the given equation. $$\csc ^{2} \theta-4=0$$

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Problem 38

(a) Graph \(f\) and \(g\) in the given viewing rectangle and find the intersection points graphically, rounded to two decimal places. (b) Find the intersection

View solution

Key Concepts

Trigonometric Identities

Trigonometric identities are essential tools in mathematics. They establish important relationships between trigonometric functions. One of the most fundamental trigonometric identities is the Pythagorean identity. This can be written as:

• \( \sin^2 x + \cos^2 x = 1 \)

This identity depicts how the squares of sine and cosine of an angle sum up to one. It forms the basis of many trigonometric calculations and problem-solving processes.

Another crucial set of identities involves half-angle identities. These identities help in finding the trigonometric functions of half of a given angle. For example:

• \( \sin \frac{x}{2} = \sqrt{\frac{1 - \cos x}{2}} \)
• \( \cos \frac{x}{2} = \sqrt{\frac{1 + \cos x}{2}} \)

These identities are especially useful when angles are not in conventional positions, like in the third or second quadrant. They require an understanding of the signs of the trigonometric functions in different quadrants.

Third Quadrant

The third quadrant of the Cartesian plane plays a vital role in trigonometry. It is defined by angles between \(180^\circ\) and \(270^\circ\). Since the angle \(x\) in the original problem falls within this range, it is situated in the third quadrant.

In this specific quadrant:

• Both sine and cosine are negative, which impacts the calculation of sine and cosine of any value of \(x\) in this range.

Knowing that cosine and sine are negative helps in evaluating trigonometric functions, especially when applied in half-angle identities.

By referring back to the quadrant details, one can confidently determine the signs of trigonometric outputs like \(\sin x = -\frac{3}{5} \) in the provided context.

Cosine Negative Angle

When dealing with trigonometric functions, understanding angles' nature like negative angles is imperative. Cosine of a negative angle, indicated as \(-\cos x\), refers to the direction of the cosine value when angles are defined on different quadrants.

In our problem, the cosine of \(x\) is \(-\frac{4}{5}\) as given because \(x\) is in the third quadrant. This reflects the inherent property that cosine functions flip their positivity or negativity according to the quadrant:

• In the first quadrant, cosine is positive.
• In the third quadrant (as in this problem), cosine values are negative.

When using cosine negative values in half-angle identities, it impacts calculations such as:

• \(\cos \frac{x}{2} = -\sqrt{\frac{1 + \cos x}{2}}\)

This ensures that the signs are correctly considered, leading to accurate evaluations.