Problem 8 - Precalculus: Functions and Graphs

Question

Find the exact values of \(\sin (\theta / 2), \cos (\theta / 2),\) and \(\tan (\theta / 2)\) for the given conditions. $$\sec \theta=-4 ; \quad 180^{\circ}<\theta<270^{\circ}$$

Step-by-Step Solution

Answer

\( \sin (\theta / 2) = \frac{\sqrt{10}}{4}, \cos (\theta / 2) = \frac{\sqrt{6}}{4}, \tan (\theta / 2) = -\frac{\sqrt{15}}{3} \).

Step 1: Identify the Quadrant for θ

Since we know that \(180^{\circ} < \theta < 270^{\circ}\), \(\theta\) lies in the third quadrant. In this quadrant, \(\cos \theta\) is negative, and \(\sin \theta\) is also negative.

Step 2: Calculate cos(θ) from sec(θ)

The secant function is the reciprocal of the cosine function. Since \(\sec \theta = -4\), we have \(\cos \theta = \frac{1}{\sec \theta} = \frac{1}{-4} = -\frac{1}{4}\).

Step 3: Determine sin(θ) Using Pythagorean Identity

Apply the Pythagorean identity \(\sin^2 \theta + \cos^2 \theta = 1\):\[\sin^2 \theta + \left(-\frac{1}{4}\right)^2 = 1\]\[\sin^2 \theta + \frac{1}{16} = 1\]\[\sin^2 \theta = 1 - \frac{1}{16} = \frac{15}{16}\]\[\sin \theta = -\sqrt{\frac{15}{16}} = -\frac{\sqrt{15}}{4}\]because \(\theta\) is in the third quadrant where sine is negative.

Step 4: Use Half-Angle Formulas

For the half-angle formulas:- \(\sin \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 - \cos \theta}{2}}\)- \(\cos \left(\frac{\theta}{2}\right) = \pm \sqrt{\frac{1 + \cos \theta}{2}}\)- \(\tan \left(\frac{\theta}{2}\right) = \frac{\sin \theta}{1 + \cos \theta}\)

Step 5: Calculate sin(θ/2)

We choose the positive value since \(\frac{\theta}{2}\) (which is between \(90^{\circ}\) and \(135^{\circ}\)) is in the second quadrant where sine is positive:\[\sin \left(\frac{\theta}{2}\right) = \sqrt{\frac{1 - \left(-\frac{1}{4}\right)}{2}} = \sqrt{\frac{1 + \frac{1}{4}}{2}} = \sqrt{\frac{\frac{5}{4}}{2}} = \sqrt{\frac{5}{8}} = \frac{\sqrt{10}}{4}\]

Step 6: Calculate cos(θ/2)

Similarly, cosine of \(\frac{\theta}{2}\) in the second quadrant is positive:\[\cos \left(\frac{\theta}{2}\right) = \sqrt{\frac{1 + \left(-\frac{1}{4}\right)}{2}} = \sqrt{\frac{1 - \frac{1}{4}}{2}} = \sqrt{\frac{\frac{3}{4}}{2}} = \sqrt{\frac{3}{8}} = \frac{\sqrt{6}}{4}\]

Step 7: Calculate tan(θ/2)

The tangent of half-angle will also be positive:\[\tan \left(\frac{\theta}{2}\right) = \frac{\sin \theta}{1 + \cos \theta} = \frac{-\frac{\sqrt{15}}{4}}{1 - \frac{1}{4}} = \frac{-\frac{\sqrt{15}}{4}}{\frac{3}{4}} = -\frac{\sqrt{15}}{3}\]

Key Concepts

Half-Angle Formulas

Half-angle formulas are special trigonometric identities that help us find the sine, cosine, and tangent values of half of a given angle. They are particularly useful when dealing with angles that are not typically found in the unit circle. The half-angle formulas are:

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

The signs of these functions depend on the quadrant where the half-angle \(\frac{\theta}{2}\) lies. This makes it essential to understand the quadrants to assign the correct signs.

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry which expresses the relationship between the squares of sine and cosine functions. It is given by the equation:

• \(\sin^2 \theta + \cos^2 \theta = 1\)

This identity helps in determining one function when the other is known. For instance, if \(\cos \theta = -\frac{1}{4}\), you can find \(\sin \theta\) using this equation. Rearranging can lead to: \[\sin^2 \theta = 1 - \cos^2 \theta\]This identity is vital for calculations involving trigonometric functions, especially when trying to simplify expressions or find unknown values.

Trigonometric Functions

Trigonometric functions include sine, cosine, tangent, secant, cosecant, and cotangent. Each function arises from the ratios of the sides of a right triangle or the unit circle. For this exercise, knowing that \(\sec \theta = -4\) allows us to find \(\cos \theta\) since the secant is the reciprocal:

• \(\sec \theta = \frac{1}{\cos \theta}\)
• \(\cos \theta = \frac{1}{-4} = -\frac{1}{4}\)

Understanding these relationships aids in solving for other functions like \(\sin \theta\) and \(\tan \theta\). Recognizing how each function relates can simplify solving trigonometric equations.

Quadrants

The coordinate plane is divided into four quadrants, each with different characteristics for trigonometric functions. • First Quadrant: Sine, cosine, and tangent are positive.• Second Quadrant: Sine is positive; cosine and tangent are negative.• Third Quadrant: Tangent is positive; sine and cosine are negative.• Fourth Quadrant: Cosine is positive; sine and tangent are negative.For \(180^{\circ} < \theta < 270^{\circ}\), \(\theta\) is in the third quadrant, where both \(\sin \theta\) and \(\cos \theta\) are negative. This understanding is crucial when determining the signs of trigonometric functions at half-angles since \(\frac{\theta}{2}\) will be located in the second quadrant, affecting the signs of \(\sin \left(\frac{\theta}{2}\right)\) and \(\cos \left(\frac{\theta}{2}\right)\).

Reciprocal Identities

Reciprocal identities connect trigonometric functions with their counterparts that "flip" the input-output relationship. The primary reciprocal identities are:

• \(\sec \theta = \frac{1}{\cos \theta}\)
• \(\csc \theta = \frac{1}{\sin \theta}\)
• \(\cot \theta = \frac{1}{\tan \theta}\)

For our scenario, knowing \(\sec \theta = -4\) translates to finding \(\cos \theta\) as its reciprocal. These identities are helpful because they provide alternative forms of expressions and solutions for trigonometric equations. They allow you to switch between functions, broadening your toolkit for solving problems efficiently.