Problem 18 - Algebra 2 and Trigonometry

Question

In \(18-27\) , express each given function value in terms of a function value of a positive acute angle (the reference angle). \(\sin 215^{\circ}\)

Step-by-Step Solution

Answer

\( \sin 215^{\circ} = -\sin(35^{\circ}) \)

Step 1: Identify the angle's quadrant

The angle is \(215^{\circ}\), which is between \(180^{\circ}\) and \(270^{\circ}\). Therefore, it lies in the third quadrant.

Step 2: Determine the reference angle

In the third quadrant, the reference angle \( \theta' \) can be found by subtracting \(180^{\circ}\) from the angle. Thus, \(\theta' = 215^{\circ} - 180^{\circ} = 35^{\circ}\).

Step 3: Relate given function to reference angle

The sine of an angle in the third quadrant is the negative of its reference angle's sine. Therefore, \( \sin 215^{\circ} = -\sin(35^{\circ})\).

Key Concepts

Trigonometric Functions

Trigonometric functions are essential tools in mathematics, particularly in geometry and calculus. They help us relate angles to the sides of triangles. The main trigonometric functions include sine (\( \sin \theta \)), cosine (\( \cos \theta \)), and tangent (\( \tan \theta \)). Each function is linked to a specific set of ratios in a right-angled triangle:

• Sine, which measures the ratio of the opposite side to the hypotenuse.
• Cosine, which gives the ratio of the adjacent side to the hypotenuse.
• Tangent, which provides the ratio of the opposite side to the adjacent side.

These functions don't just serve triangles; they extend to circles through the unit circle concept. Extending angles beyond 0° to 360° involves these functions making trigonometry applicable in various fields like physics, engineering, and even art.
Understanding these functions is key to exploring complex topics like periodic functions, waves, and oscillations. They also hold immense value because we can cycle through functions to learn about transformations, translations, and scaling in a broader scope of mathematics.

Sine Function

The sine function, represented as \( \sin \theta \), is one of the primary trigonometric functions. Its main job is to connect the angle of a right triangle to the length of its opposite side divided by its hypotenuse. When drawing a triangle within the unit circle, where the circle's radius equals 1, the sine of an angle is the y-coordinate of the corresponding point on the circle.
This property makes sine wonderful for modeling oscillating behaviors such as sound waves or pendulum swings. The sine function is periodic with a period of \( \360^{\circ} \) (or \( 2\pi \) radians) because its values repeat every complete circle:

• The sine of 0° is 0.
• The sine of 90° is 1.
• The sine of 180° reverts back to 0.
• The sine of 270° is -1.
• The sine of 360° caps off the cycle back at 0.

Sine's periodic nature gives it an endless cycle of ups and downs, vital for understanding wave cycles and harmonics in musical tunes or signals.

Angle Quadrants

In trigonometry, the concept of angle quadrants is crucial for understanding how angle measurements translate across different ranges. The four quadrants on a coordinate plane determine how the signs of trigonometric functions play out.
They section off the plane into four parts, each with distinct properties:

• **First Quadrant**: This is where all angles from 0° to 90° reside. Trigonometric functions are positive here.
• **Second Quadrant**: Between 90° and 180°, sine remains positive, but cosine and tangent turn negative.
• **Third Quadrant**: Spanning 180° to 270°, tangent emerges as the positive function, while sine and cosine go negative.
• **Fourth Quadrant**: Here (270° to 360°), cosine stands positive, but sine and tangent are negative.

Understanding which trigonometric functions are positive or negative in each quadrant aids us in simplifying angle calculations, like determining reference angles. Knowing how signs alternate is especially useful in physics and engineering, where directional aspects align with trigonometric values positivity or negativity.