Problem 49 - Algebra 2

Question

Open-Ended Choose an angle measure \(A .\) a. Find \(\sin A\) and \(\cos A .\) b. Use an identity to find \(\sin 2 A\) c. Use an identity to find \(\cos \frac{A}{2}\)

Step-by-Step Solution

Answer

For \(A = 30^\circ\), \(\sin A = 0.5\), \(\cos A = \sqrt{3}/2\), \(\sin 2A = \sqrt{3}/2\), \(\cos A/2 = \pm \sqrt{3 + 2 \sqrt{3}} / 4\)

Step 1: Choose Angle

Let's choose an angle for \(A\). For simplicity, we will use \(A = 30^\circ\). This angle is frequently used in trigonometry and its sine and cosine values are widely recognized.

Step 2: Finding Sin and Cos

Calculate \(\sin A\) and \(\cos A\). For \(A = 30^\circ\), \(\sin A = 0.5\) and \(\cos A = \sqrt{3}/2\) because this is a commonly used angle and we know its sine and cosine values.

Step 3: Finding Sin2A

To find \(\sin 2A\), use the double angle identity \(\sin 2A = 2 \sin A \cos A\). Substitute \(\sin A = 0.5\) and \(\cos A = \sqrt{3}/2\) into the equation gives \(\sin 2A = 2 * 0.5 * \sqrt{3}/2 = \sqrt{3}/2\).

Step 4: Finding CosA/2

To find \(\cos A/2\), use the half angle identity \(\cos A/2 = \pm \sqrt{(1 + \cos A) / 2}\). Substitute \(\cos A = \sqrt{3}/2\) into the equation gives \(\cos A/2 = \pm \sqrt{(1 + \sqrt{3}/2) / 2} = \pm \sqrt{3 + 2 \sqrt{3}} / 4\) . Keep in mind, the plus or minus sign is decided based on the quadrant where angle \(A/2\) lies.

Key Concepts

Sine Function

The sine function is one of the basic trigonometric functions, commonly denoted as \( \sin \theta \), where \( \theta \) represents an angle. The function helps us relate the angle of a right triangle to the ratio of the length of the side opposite the angle to the length of the triangle's hypotenuse. This is particularly useful in solving geometric and physics problems involving right triangles.
For example, if we choose the angle \( A = 30^\circ \), as in our exercise, the sine value is often memorized or found using sine tables: \( \sin 30^\circ = 0.5 \).
It’s important to remember that the sine function is periodic with a period of \( 360^\circ \) or \( 2\pi \) radians. This means it repeats its values every \( 360^\circ \). The range of the sine function is between -1 and 1, meaning regardless of the angle, \( \sin \theta \) will always lie within this range.

Cosine Function

The cosine function, denoted as \( \cos \theta \), is another fundamental function used to relate an angle of a triangle to the lengths of the sides within the triangle. Specifically, it is the ratio of the length of the adjacent side to that of the hypotenuse.
Following our example for \( A = 30^\circ \), the cosine value can be found using trigonometric identities or memorized values: \( \cos 30^\circ = \sqrt{3}/2 \). This value is commonly used in various mathematical applications, from simple geometry to complex calculus.
The cosine function also has a period of \( 360^\circ \) or \( 2\pi \), similar to the sine function. However, its graph is a **phase-shifted** version of the sine function and starts at 1 when \( \theta = 0 \). Like |**sine**|, cosine’s range is between -1 and 1.
Understanding these properties allows the cosine function to be a powerful tool in analyzing cyclic phenomena, such as sound waves, alternating current circuits, and many more applications in physics.

Double Angle Identities

Double angle identities are crucial for simplifying expressions where angles are doubled. Each trigonometric function, like sine and cosine, has its own double angle identity.
Specifically, for the sine function, the double angle identity is \( \sin 2A = 2 \sin A \cos A \). By using this identity, you can calculate \( \sin 2A \) easily if you know \( \sin A \) and \( \cos A \).
With our chosen angle \( A = 30^\circ \), applying the double angle identity gives \( \sin 2 \times 30^\circ = 2 \times \sin 30^\circ \times \cos 30^\circ = 2 \times 0.5 \times \frac{\sqrt{3}}{2} = \sqrt{3}/2 \).
These identities help simplify and solve trigonometric equations and are widely used in integration and differentiation, finding the values of trigonometric functions at special angles, and in simplifying trigonometric expressions in equations.

Half Angle Identities

Half angle identities are used to find the values of trigonometric functions when the angle is halved, which simplifies the calculations involved in complex trigonometric expressions.
One of the half angle identities is for the cosine function: \( \cos \frac{A}{2} = \pm \sqrt{(1 + \cos A)/2} \). This formula allows us to find the cosine of half of a given angle based on its known cosine value.
Using our chosen angle \( A = 30^\circ \), the corresponding calculations would be \( \cos 15^\circ = \pm \sqrt{(1 + \cos 30^\circ) / 2} = \pm \sqrt{(1 + \sqrt{3}/2) / 2} = \pm \sqrt{\frac{3 + \sqrt{3}}{4}} \).
The sign outcome of \( \pm \) is based on the quadrant where the half-angle lies. For precise results, understanding these quadrants is crucial.
Half angle identities not only aid in finding trigonometric values of difficult angles but also assist in integration and derivation in calculus by transforming complex expressions into manageable ones.