This week in math we learned how to find exact trigonometric ratios when given an exact point on the terminal arm or a single trignometric ratio. This skills is important cause it is great for encreasing spatial awerness and can be applied to other subjects like physics. I choose this subject because of how little information is needed to find a ton of values.

1. Finding the Exact Trigonometric Ratios for Rotation Angles from 0° to 360° Given a Point on the Terminal Arm

Steps:

1. Identify the Coordinates: Determine the coordinates (x, y) of the point on the terminal arm.

2. Calculate the Radius (r): Use the formula to find the radius (hypotenuse).

3. Find the Trigonometric Ratios:
– Sine (sin θ):
– Cosine (cos θ):
– Tangent (tan θ):

Example:

Given the point (3, 4) on the terminal arm:
–
–
–

Trigonometric Ratios:
–
–
–

2. Determining Trigonometric Ratios for a Rotation Angle from 0° to 360° Given a Different Trigonometric Ratio for the Angle

Steps:

1. Identify the Given Ratio: Determine which trigonometric ratio is provided (e.g., sin, cos, tan).

2.Determine the Quadrants: Based on the given ratio’s sign, determine the possible quadrants where the angle could lie.
– Sine (sin θ):
– Positive in Quadrants I and II
– Negative in Quadrants III and IV
– Cosine (cos θ):
– Positive in Quadrants I and IV
– Negative in Quadrants II and III
– Tangent (tan θ):
– Positive in Quadrants I and III
– Negative in Quadrants II and IV

3. Use Reference Angles: Find the reference angle (θ) using the inverse function of the given ratio.

4. Determine the Actual Angles: Add or subtract the reference angle from 180° or 360° based on the quadrants identified.

Example:

Given :

– Since is positive, θ could be in Quadrant I or II.
– Reference angle .

Actual angles:
– In Quadrant I:
– In Quadrant II:

To find other trigonometric ratios at these angles:
– For :
–
–
–

– For
–

–
–

Using these steps and examples, you can determine the exact trigonometric ratios for any given point or provided ratio.