Problem 20 - Algebra 2 and Trigonometry

Question

In \(3-22 :\) a. Rewrite each function value in terms of its cofunction. b. Find, to four decimal places, the value of the function value found in a. $$ \sin 295^{\circ} $$

Step-by-Step Solution

Answer

The value of \( \sin 295^{\circ} \) is approximately \(-0.9063\).

Step 1: Understanding the Cofunction Identity

The cofunction identity for sine and cosine is \(\sin(90^{\circ} - x) = \cos(x)\). This means sine of an angle is equal to cosine of its complementary angle. However, when the angle is expressed in degrees greater than 90, we need to find an equivalent angle through cofunction identities and symmetric properties.

Step 2: Convert Angle using the Cofunction Property

To find the cofunction, first convert \(\sin 295^{\circ}\) to an angle we can use this identity on. For sine, it's straightforward: because sine has a period of 360 degrees, we can use the symmetric property to find:\( \sin 295^{\circ} = \sin (360^{\circ} - 295^{\circ}) = -\sin 65^{\circ} \). This gives us \( -\cos 25^{\circ} \) using the cofunction identity \( \sin 65^{\circ} = \cos(90^{\circ} - 65^{\circ}) = \cos 25^{\circ} \) and considering the negative due to quadrant rules.

Step 3: Evaluate the Cofunction

Now calculate \( -\cos 25^{\circ} \) using a calculator, as required for precise values. Ensure that the calculator is in degree mode, as the angle is in degrees. Calculating yields approximately \(-0.9063\).

Key Concepts

Sine and Cosine Relationships

In trigonometry, one of the key ideas is that sine and cosine are related through complementary angles. This relationship is known as the cofunction identity, which tells us that the sine of any angle is equal to the cosine of its complementary angle and vice versa. So, we have

• \( \sin(90^{\circ} - x) = \cos(x) \)
• \( \cos(90^{\circ} - x) = \sin(x) \)

This identity is helpful because it allows us to switch between sine and cosine when dealing with specific angles. Understanding this relationship aids in simplifying expressions and solving trigonometric equations. It's like knowing that two keys open the same door, depending on the angle you're viewing it from.

Symmetric Properties of Trigonometric Functions

Trigonometric functions exhibit symmetric properties that help us evaluate angles when they exceed certain limits. We know sine and cosine are periodic with a cycle of 360 degrees. This means that trigonometric values repeat over this interval. Additionally, sine has an odd symmetry, meaning \( \sin(-x) = -\sin(x) \).
Similarly, cosine has an even symmetry: \( \cos(-x) = \cos(x) \). For angles like 295 degrees, which is in the fourth quadrant, sine becomes negative, since

• \( \sin(360^{\circ} - x) = -\sin(x) \)
• \( \cos(360^{\circ} - x) = \cos(x) \)

These symmetric properties simplify the process of finding equivalent angles and solving complex trigonometric problems. They ensure that even when angles are not conventional, we can calculate their sine or cosine by relating them back to a value in the first quadrant, where the functions are easiest to handle.

Angle Conversion in Trigonometry

In trigonometry, converting angles is sometimes necessary, especially when working with angles greater than 90 degrees or less than 0 degrees. Conversion helps in rewriting trigonometric functions to make use of identities or when evaluating with a calculator. Angles can be converted using their periodic nature:

• A full circle is 360 degrees, meaning any angle of the form \( x + 360^{\circ} \) returns the same trigonometric function value as \( x^{\circ} \).
• Complementary angles add up to 90 degrees, while supplementary angles add up to 180 degrees.

Using these properties, the angle can be adjusted to find an equivalent angle within the 0 to 90 degrees range. For example, in the exercise, we adjusted \( \sin 295^{\circ} \) using the property of periodicity and symmetry to \( -\sin 65^{\circ} \), simplifying calculations. This way of thinking helps us stay within the more straightforward first quadrant as much as possible, making the math much easier to grasp.