Problem 58 - Algebra and Trigonometry

Question

Use the given information to find the exact value of each of the following: a. \(\sin \frac{\alpha}{2}\) b. \(\cos \frac{\alpha}{2}\) c. \(\tan \frac{\alpha}{2}\) $$ \sec \alpha=-3, \frac{\pi}{2}<\alpha<\pi $$

Step-by-Step Solution

Answer

The exact values are \(\sin \frac{\alpha}{2} = \sqrt{\frac{2}{3}}\), \(\cos \frac{\alpha}{2} = \frac{1}{\sqrt{3}}\), and \(\tan \frac{\alpha}{2} = \sqrt{2}\).

Step 1: Get the Value for Cosine from Secant

The secant of an angle is the reciprocal of the cosine of that angle. With the help of this information, it can be said that \(\cos \alpha = -\frac{1}{3}\).

Step 2: Calculate Sin of Angle

The sine of \(\alpha\) is given by the Pythagorean identity \(\sin^2 \alpha + \cos^2 \alpha = 1\). Solving for \(\sin \alpha\), there will be two possibilities: \(\sqrt{1- \frac{1}{9}}\) and \(-\sqrt{1- \frac{1}{9}}\). Since \(\frac{\pi}{2}<\alpha<\pi\) which is in the second quadrant, only positive root is accepted. So, \(\sin \alpha = \sqrt{\frac{8}{9}} = \frac{2\sqrt{2}}{3}\).

Step 3: Use Half-Angle Formulas for Sin and Cos

The half-angle formulas for sine and cosine are as follows: \(\sin \frac{\alpha}{2} = ±\sqrt{\frac{1-\cos \alpha}{2}}\) and \(\cos \frac{\alpha}{2} = ±\sqrt{\frac{1+\cos \alpha}{2}}\). Again, since \(\alpha\) is in second quadrant (\(\frac{\pi}{2}<\alpha<\pi\)), \(\frac{\alpha}{2}\) would be in the first quadrant (\(0<\frac{\alpha}{2}<\frac{\pi}{2}\)) where sine and cosine are positive. Therefore, \(\sin \frac{\alpha}{2} = \sqrt{\frac{1-\left(-\frac{1}{3}\right)}{2}} = \sqrt{\frac{2}{3}}\) and \(\cos \frac{\alpha}{2} = \sqrt{\frac{1+\left(-\frac{1}{3}\right)}{2}} = \frac{1}{\sqrt{3}}\).

Step 4: Use Quotient Identity for Tan

The value of \(\tan \frac{\alpha}{2}\) is obtained using the quotient identity which is given by \(\tan \alpha=\frac{\sin \alpha}{\cos \alpha}\). Therefore, \(\tan \frac{\alpha}{2} = \frac{\sin \frac{\alpha}{2}}{\cos \frac{\alpha}{2}} = \sqrt{2}\).

Key Concepts

Pythagorean identity

The Pythagorean identity is a fundamental aspect of trigonometry and it states that for any angle, the square of the sine plus the square of the cosine equals one: \[ \sin^2(\theta) + \cos^2(\theta) = 1 \. \] This relationship is derived from the Pythagorean theorem applied to a unit circle, where the hypotenuse is always 1. In our exercise, this identity was used to find the sine of the angle \( \alpha \) given the cosine, which was determined from the given secant value. Understanding how to use this identity is crucial because it can help solve many trigonometric problems, especially when we know one trigonometric function and need to find another.

Improvement advice for students would be to remember that the Pythagorean identity can also be rearranged to isolate either \( \sin \theta \) or \( \cos \theta \) when only the other is known. Additionally, understanding that this identity holds for all angles can empower students to deal with various trigonometric problems.

Trigonometric functions

Trigonometric functions are the backbone of trigonometry and include sine, cosine, tangent, along with their reciprocals: cosecant, secant, and cotangent. These functions help relate the angles of a triangle to the lengths of its sides and are defined initially for acute angles in a right triangle but can be extended to angles of any size.

In our exercise, we applied half-angle formulas, which are specific cases of trigonometric functions for \( \frac{\alpha}{2} \) when \( \alpha \) is known. These formulas are \[ \sin \frac{\alpha}{2} = \pm\sqrt{\frac{1-\cos \alpha}{2}} \. \] and \[ \cos \frac{\alpha}{2} = \pm\sqrt{\frac{1+\cos \alpha}{2}} \. \] The signs depend on the quadrant of the angle \( \frac{\alpha}{2} \) lies in. For students working to improve, it's beneficial to focus on memorizing these fundamental formulas and understanding how the quadrant of an angle affects the signs of the trigonometric functions.

Second quadrant angles

Angles in the second quadrant range between \( \frac{\pi}{2} \) and \( \pi \) radians. In this region of the coordinate system, the sine of an angle is positive, while the cosine and tangent are negative. This characteristic is vital when determining the values of trigonometric functions for angles within this quadrant, as was important in our exercise.

When solving for \( \sin \alpha \) after determining \( \cos \alpha \) from the secant value, we found two possibilities due to the square root but chose the positive one, consistent with the sign convention for the second quadrant. Students should note the need for understanding these quadrant-related sign rules to determine the correct values of trigonometric functions. Remembering the acronym 'All Students Take Calculus' can help recall which functions are positive in each quadrant, beginning with All (all positive) in the first quadrant and proceeding counterclockwise with Students (sine positive) in the second quadrant.