Problem 36 - Algebra and Trigonometry

Question

35-40 Find \(\sin \frac{x}{2}, \cos \frac{x}{2},\) and \(\tan \frac{x}{2}\) from the given information. \(\cos x=-\frac{4}{5}, \quad 180^{\circ} < x < 270^{\circ}\)

Step-by-Step Solution

Answer

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

Step 1: Determine the value of \( \sin x \)

Since \( x \) is in the third quadrant, where both sine and cosine are negative, use the Pythagorean identity: \( \sin^2 x + \cos^2 x = 1 \). Substitute the given value of \( \cos x = -\frac{4}{5} \):\[ \sin^2 x + \left(-\frac{4}{5}\right)^2 = 1 \] \[ \sin^2 x + \frac{16}{25} = 1 \] \[ \sin^2 x = \frac{9}{25} \]. \( \sin x \) is negative in the third quadrant, so \( \sin x = -\frac{3}{5} \).

Step 2: Calculate \( \sin \frac{x}{2} \) using the half-angle formula

The formula for \( \sin \frac{x}{2} \) is \( \pm \sqrt{\frac{1 - \cos x}{2}} \). Substitute \( \cos x = -\frac{4}{5} \): \[ \sin \frac{x}{2} = \pm \sqrt{\frac{1 + \frac{4}{5}}{2}} = \pm \sqrt{\frac{9}{10}} = \pm \frac{3\sqrt{10}}{10} \]. Since \( \frac{x}{2} \) should meet the criteria of \( 90^{\circ} < \frac{x}{2} < 135^{\circ} \), \( \sin \frac{x}{2} \) is positive, so \( \sin \frac{x}{2} = \frac{3\sqrt{10}}{10} \).

Step 3: Calculate \( \cos \frac{x}{2} \) using the half-angle formula

The formula for \( \cos \frac{x}{2} \) is \( \pm \sqrt{\frac{1 + \cos x}{2}} \). Substitute \( \cos x = -\frac{4}{5} \): \[ \cos \frac{x}{2} = \pm \sqrt{\frac{1 - \frac{4}{5}}{2}} = \pm \sqrt{\frac{1}{10}} = \pm \frac{1}{\sqrt{10}} = \pm \frac{\sqrt{10}}{10} \]. Since \( \frac{x}{2} \) is in the second quadrant, where the cosine is negative, \( \cos \frac{x}{2} = -\frac{\sqrt{10}}{10} \).

Step 4: Calculate \( \tan \frac{x}{2} \) using sine and cosine half-angle values

Use the identity \( \tan \frac{x}{2} = \frac{\sin \frac{x}{2}}{\cos \frac{x}{2}} \). Substitute the previously found values: \[ \tan \frac{x}{2} = \frac{\frac{3\sqrt{10}}{10}}{-\frac{\sqrt{10}}{10}} \]. Simplifying, \( \tan \frac{x}{2} = -3 \).

Key Concepts

Third Quadrant

The unit circle can be divided into four quadrants, each representing a specific range of angles and corresponding sign values for trigonometric functions like sine, cosine, and tangent.
In the third quadrant, angles range from 180° to 270°. What's unique about this quadrant is:

• The sine and cosine values are negative. This is because both the x-coordinate (cosine) and the y-coordinate (sine) in this quadrant are below their respective axes.
• The tangent, derived from the ratio of sine to cosine, is positive, essentially because a negative divided by a negative yields a positive result.

Understanding the third quadrant's traits is fundamental as it helps determine the signs of angle specific values without further calculation. Especially useful for problems reliant on half-angle formulas.

Half-Angle Formulas

Half-angle formulas are vital trigonometric identities that let us find the sine, cosine, and tangent of half an angle when the values of the whole angle are known. These formulas derive from the double-angle identities and can be expressed as:

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

These identities are particularly useful in calculating trigonometric values for angles that aren't typically listed on a standard trigonometric table by using the known values of related angles. The choice of the positive or negative sign depends not only on the angle but also on its specific quadrant location, which dictates the trigonometric function's behavior.Understanding these formulas gives us the tools to smoothly transition between angular measures, a common requirement in many mathematical and physics problems.

Pythagorean Identity

The Pythagorean identity is an essential building block in trigonometry, closely related to the unit circle. It states: \[ \sin^2 x + \cos^2 x = 1 \]This identity is derived from the Pythagorean theorem and highlights the relationship between the sine and cosine of an angle.

• If one of the values (\( \sin x \) or \( \cos x \)) is known, the other can be easily determined using this identity.
• It also serves as the foundation for many other trigonometric identities and equations.

For angles located in different quadrants, including the third quadrant, the identity remains true. However, the signs of the trigonometric functions will vary according to the quadrant in which the angle is located. As a result, whenever faced with a problem on trigonometric identities, especially involving unknown sine or cosine values, the Pythagorean identity proves indispensable in resolving them.