Function Values in Trigonometry- Evaluating Trig Functions
What Are Trig Function Values, Anyway?
Trigonometry throws a lot of people off when they hit function values. You're expected to find the sine, cosine, and tangent of angles—and suddenly you're staring at a circle covered in numbers that don't make sense.
Here's the brutal truth: trig function values are just ratios. That's it. They compare sides of a right triangle. Once that clicks, evaluating them becomes mechanical.
The Three Functions You Actually Need
Most problems only care about three trig functions:
- Sine (sin) = opposite side ÷ hypotenuse
- Cosine (cos) = adjacent side ÷ hypotenuse
- Tangent (tan) = opposite side ÷ adjacent side
The other three—cosecant, secant, and cotangent—are just reciprocals. If you know the big three, you can figure out the rest.
The Unit Circle: Your Best Friend
Forget memorizing random numbers. The unit circle gives you every trig value you'll ever need for the standard angles.
Draw a circle with radius 1 centered at the origin. Any point on this circle has coordinates (cos θ, sin θ). The x-coordinate is cosine. The y-coordinate is sine. That's why sin²θ + cos²θ = 1—it's just the Pythagorean theorem on a unit circle.
Values for the Standard Angles
These angles show up constantly: 0°, 30°, 45°, 60°, and 90°. Memorize this table or you will struggle forever.
| Angle | sin | cos | tan |
|---|---|---|---|
| 0° | 0 | 1 | 0 |
| 30° | 1/2 | √3/2 | 1/√3 |
| 45° | √2/2 | √2/2 | 1 |
| 60° | √3/2 | 1/2 | √3 |
| 90° | 1 | 0 | undefined |
Notice the pattern: sine increases as you go down the table, cosine decreases. Tangent is just sin ÷ cos.
Reference Angles: The Shortcut
What about angles like 150° or 210°? You don't start from scratch. A reference angle is the acute angle between your angle and the x-axis.
Here's how to find trig values for any angle:
- Find the reference angle (subtract from 180° if in QII, subtract from 360° if in QIII, etc.)
- Get the trig value for that reference angle from your table
- Apply the sign based on the quadrant—positive or negative
Quadrant Rules
- Quadrant I: All functions positive
- Quadrant II: Only sine positive
- Quadrant III: Only tangent positive
- Quadrant IV: Only cosine positive
How to Actually Evaluate Trig Functions
Let's work through a real example. Find sin(150°).
- 150° is in Quadrant II
- Reference angle = 180° - 150° = 30°
- sin(30°) = 1/2 (from the table)
- Sine is positive in Quadrant II
- Answer: sin(150°) = 1/2
Try cos(225°) now:
- 225° is in Quadrant III
- Reference angle = 225° - 180° = 45°
- cos(45°) = √2/2
- Cosine is negative in Quadrant III
- Answer: cos(225°) = -√2/2
When to Use Your Calculator
The table covers standard angles. Everything else requires a calculator—and you need to know which mode you're in.
- Set it to DEG for degrees
- Set it to RAD for radians
Using degrees when your calculator is in radian mode gives you garbage answers. This trips up more students than almost anything else.
Common Mistakes That Cost You Points
- Mixing up sin and cos: X-axis = cosine, Y-axis = sine. Don't forget it.
- Forgetting the sign: A positive value in the wrong quadrant is still wrong.
- Tangent at 90°: It's undefined. The line never intersects. Move on.
- Rounding too early: Keep fractions exact until the final answer.
Quick Reference: Reciprocal Functions
If a problem asks for csc, sec, or cot—just flip the basic function:
- csc θ = 1/sin θ
- sec θ = 1/cos θ
- cot θ = 1/tan θ = cos θ/sin θ
So if sin(30°) = 1/2, then csc(30°) = 2. No extra memorization needed.
Bottom Line
Evaluating trig functions comes down to three things:
- Know the unit circle and the standard angle values
- Understand reference angles and quadrant signs
- Keep your calculator in the right mode
Drill these until they're automatic. The problems will vary, but the process never does.