The six trigonometric functions connect angles to ratios, coordinates, and periodic motion. They are essential in geometry, physics, engineering, computer graphics, and any situation involving circles or waves. A unit circle gives a clean way to define these functions for every angle, not just the acute angles of a right triangle.
By using one circle, you can see values, signs, and relationships all at once.
Key Facts
- On the unit circle, a point at angle θ has coordinates (x, y) = (cos θ, sin θ).
- sin θ = opposite/hypotenuse and cos θ = adjacent/hypotenuse.
- tan θ = sin θ/cos θ = y/x, when x ≠ 0.
- csc θ = 1/sin θ, sec θ = 1/cos θ, and cot θ = 1/tan θ.
- Pythagorean identity: sin² θ + cos² θ = 1.
- Quadrant signs: QI all positive, QII sin and csc positive, QIII tan and cot positive, QIV cos and sec positive.
Vocabulary
- Sine
- Sine of an angle is the y-coordinate on the unit circle or the ratio opposite/hypotenuse in a right triangle.
- Cosine
- Cosine of an angle is the x-coordinate on the unit circle or the ratio adjacent/hypotenuse in a right triangle.
- Tangent
- Tangent of an angle is the ratio sin θ/cos θ or opposite/adjacent when the denominator is not zero.
- Reciprocal function
- A reciprocal trigonometric function is formed by taking 1 divided by a basic trig function, such as sec θ = 1/cos θ.
- Unit circle
- The unit circle is the circle centered at the origin with radius 1, used to define trig functions for any angle.
Common Mistakes to Avoid
- Swapping sine and cosine, which gives the wrong coordinate or triangle side ratio. On the unit circle, cos θ is x and sin θ is y.
- Forgetting quadrant signs, which makes answers positive when they should be negative. Always locate the angle's quadrant before assigning a sign.
- Treating tangent as defined everywhere, which is wrong when cos θ = 0. Tangent is undefined at angles such as 90° and 270° because division by zero is not allowed.
- Confusing reciprocal functions with inverse trig functions, which changes the meaning completely. csc θ means 1/sin θ, while sin⁻¹ θ means an angle whose sine is θ.
Practice Questions
- 1 An angle θ on the unit circle has point (3/5, 4/5). Find sin θ, cos θ, tan θ, sec θ, csc θ, and cot θ.
- 2 For θ = 210°, determine the quadrant and the signs of sin θ, cos θ, tan θ, sec θ, csc θ, and cot θ.
- 3 Explain why sin² θ + cos² θ = 1 follows from the unit circle definition of sine and cosine.