Problem 40 - Algebra and Trigonometry

Question

35-40 Find \(\sin \frac{x}{2}, \cos \frac{x}{2},\) and \(\tan \frac{x}{2}\) from the given information. \(\cot x=5, \quad 180^{\circ} < x < 270^{\circ}\)

Step-by-Step Solution

Answer

\( \sin \frac{x}{2} > 0, \cos \frac{x}{2} < 0, \tan \frac{x}{2} < 0 \).

Step 1: Understand the Given Information

We are given that \( \cot x = 5 \) and the angle \( x \) is in the third quadrant where \( 180^{\circ} < x < 270^{\circ} \).

Step 2: Use Identity for Cotangent

Recall the identity for cotangent: \( \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} = 5 \). This implies \( \frac{\cos x}{\sin x} = 5 \) which gives us \( \cos x = 5 \sin x \).

Step 3: Determine Signs of Sine and Cosine

In the third quadrant, both sine and cosine are negative. Hence, \( \sin x < 0 \) and \( \cos x < 0 \).

Step 4: Use Pythagorean Identity

Use \( \sin^2 x + \cos^2 x = 1 \). Substitute \( \cos x = 5 \sin x \) to get \( \sin^2 x + (5 \sin x)^2 = 1 \).

Step 5: Solve for Sine

Simplify to get \( \sin^2 x + 25 \sin^2 x = 1 \) or \( 26 \sin^2 x = 1 \). Solve for \( \sin x \) to find \( \sin x = -\sqrt{\frac{1}{26}} \).

Step 6: Solve for Cosine

Since \( \cos x = 5 \sin x \), substitute the value of \( \sin x \) to find \( \cos x = 5 \times -\sqrt{\frac{1}{26}} = -5\sqrt{\frac{1}{26}} \).

Step 7: Use Half-Angle Formulas

Use the half-angle formulas:\[ \sin \frac{x}{2} = \pm \sqrt{\frac{1 - \cos x}{2}}, \quad \cos \frac{x}{2} = \pm \sqrt{\frac{1 + \cos x}{2}}, \quad \tan \frac{x}{2} = \frac{\sin x}{1 + \cos x}. \]

Step 8: Evaluate Half-Angle Sine

\( \cos x = -5\sqrt{\frac{1}{26}} \), so \( 1 - \cos x = 1 + 5\sqrt{\frac{1}{26}} \).Evaluate \( \sin \frac{x}{2} = \pm \sqrt{\frac{1 + 5\sqrt{\frac{1}{26}}}{2}}. \)

Step 9: Evaluate Half-Angle Cosine

Evaluate \( \cos \frac{x}{2} = \pm \sqrt{\frac{1 - 5\sqrt{\frac{1}{26}}}{2}}. \)

Step 10: Evaluate Half-Angle Tangent

Substitute the half-angle value for sine and cosine into the tangent formula to get:\[ \tan \frac{x}{2} = \frac{-\sqrt{\frac{1}{26}}}{1 + -5\sqrt{\frac{1}{26}}}. \]Simplify to find \( \tan \frac{x}{2} \).

Step 11: Determine Correct Signs for Half-Angles

Since \( x \) is in the third quadrant, \( \frac{x}{2} \) is in the second quadrant where the sine is positive, cosine is negative, and tangent is also negative. Choose appropriate signs for half-angles.

Key Concepts

Cotangent

Cotangent, abbreviated as "cot," is one of the main trigonometric functions. It is closely related to the tangent function, and its definition is:

• \( \cot x = \frac{1}{\tan x} = \frac{\cos x}{\sin x} \)

This means cotangent is the ratio of the adjacent side to the opposite side in a right triangle. In our example problem, we know \( \cot x = 5 \), meaning that the cosine of the angle is five times the sine of the angle. Understanding this relationship allows us to explore further into other identities and calculations involving the angle.
When interpreting the cotangent, it's crucial to consider the quadrant due to signs of sine and cosine, ensuring accurate results.

Pythagorean Identity

The Pythagorean Identity is foundational in trigonometry. It states that for any angle \( x \),

• \( \sin^2 x + \cos^2 x = 1 \)

This identity allows you to find one trigonometric function if you have the other. For instance, if you know \( \sin x \), you can find \( \cos x \) via:

• \( \cos^2 x = 1 - \sin^2 x \)

In this exercise, we used \( \cos x = 5 \sin x \) in the Pythagorean Identity, leading to the equation \( \sin^2 x (1 + 25) = 1 \). By solving for \( \sin x \), we get \( \sin x = -\sqrt{\frac{1}{26}} \), considering the negative sign since \( x \) is in the third quadrant.

Half-Angle Formulas

Half-angle formulas are useful for finding trigonometric values of half-angles. These formulas are derived from the double-angle identities, and include:

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

For the given problem, these formulas allow us to determine \( \sin \frac{x}{2} \), \( \cos \frac{x}{2} \), and \( \tan \frac{x}{2} \). Using the computed values of \( \cos x \) and \( \sin x \), ensure that when choosing the sign, you consider the quadrant of the half-angle, which affects the sign positively or negatively.

Third Quadrant

The third quadrant of the unit circle is where both sine and cosine have negative values. This is the range \( 180^{\circ} < x < 270^{\circ} \). It’s important to understand the signs here because they affect the trigonometric function values:

• Sine (\( \sin x \)) is negative.
• Cosine (\( \cos x \)) is negative.
• Tangent (\( \tan x \)) becomes positive, as it is the ratio \( \frac{\sin x}{\cos x} \).

In our exercise, since \( x \) is in the third quadrant, both \( \sin x \) and \( \cos x \) are negative, which is crucial when applying the Pythagorean Identity and determining values using half-angle formulas. This quadrant's specific characteristics guide us in selecting the correct signs for trigonometric calculations.

Trigonometric Functions

Trigonometric functions describe relationships among the angles and sides of triangles. The main functions include sine (\( \sin \)), cosine (\( \cos \)), tangent (\( \tan \)), cosecant (\( \csc \)), secant (\( \sec \)), and cotangent (\( \cot \)). These functions help solve a variety of problems involving periodic phenomena and angles.
Each function has a specific relationship:

• Sine is the opposite/hypotenuse ratio.
• Cosine is the adjacent/hypotenuse ratio.
• Tangent is the opposite/adjacent ratio.

Recognizing these relationships is key in geometry and physics, where they help model waves, oscillations, and circular motions. Learning to derive one from another, using identities like the Pythagorean Identity, aids in solving trigonometric equations, such as our current exercise.