AP® Precalculus review sheet from Aim for Five (aimforfive.com/precalc/units/3/3-2)
Unit 3 · Topic 3.2
3.2 Sine, Cosine, and Tangent
Sine, cosine and tangent are defined using an angle in standard position and the point where its terminal ray meets a circle. On the unit circle, cosine is the x-coordinate, sine is the y-coordinate, and tangent is the slope of the ray. Angles are usually measured in radians.
Key terms
- unit circle
- radian
- standard position
- terminal ray
- sine, cosine, tangent
Angles in standard position
An angle is in standard position when its vertex is at the origin and its starting side (the initial ray) lies along the positive x-axis. The other side is the terminal ray.
Positive angles rotate counterclockwise from the positive x-axis. Negative angles rotate clockwise.
Angles that share a terminal ray are called coterminal. They differ by a whole number of full turns: θ, θ + 2π, θ − 2π, and so on (or θ ± 360° in degrees).
Radian measure
Draw a circle of radius r centered at the vertex. The radian measure of the angle is the length of the arc it cuts off divided by the radius: θ = s/r. On the unit circle (radius 1), the angle in radians is simply the arc length.
So 1 radian is the angle that cuts off an arc exactly as long as the radius, a little over 57°. A full turn cuts off the whole circumference, 2πr, so a full turn is 2π radians.
To convert, use 180° = π radians. Multiply degrees by π/180 to get radians, and multiply radians by 180/π to get degrees.
Rearranged, s = rθ gives arc length, but only when θ is in radians.
Sine and cosine
Draw a circle of radius r centered at the origin, and call P = (x, y) the point where the terminal ray crosses it. Then sin θ = y/r: how far P sits above or below the x-axis, divided by P's distance from the origin. And cos θ = x/r: how far P sits right or left of the y-axis, divided by that same distance.
On the unit circle, r = 1, so sin θ is just the y-coordinate of P and cos θ is just the x-coordinate. That's the picture to keep in your head: P = (cos θ, sin θ).
The ratios don't depend on which circle you use. A bigger circle gives a bigger point, but the same ratios.
Tangent
tan θ is the slope of the terminal ray. Slope is rise over run, and any point on the ray gives the same slope, so tan θ = y/x = sin θ / cos θ.
When the terminal ray is vertical, x = 0, the slope is undefined, and so is tan θ. That happens at θ = π/2, 3π/2 and their coterminal angles.
| Quadrant | sin θ | cos θ | tan θ |
|---|---|---|---|
| I (0 < θ < π/2) | + | + | + |
| II (π/2 < θ < π) | + | − | − |
| III (π < θ < 3π/2) | − | − | + |
| IV (3π/2 < θ < 2π) | − | + | − |
Worked examples
Try each one yourself first, then open the solution.
- Example 1
Converting and coterminal angles
(a) Convert 150° to radians. (b) Convert −π/4 to degrees. (c) Find the angle between 0 and 2π that is coterminal with 17π/6.
Show the solutionHide the solution
- Step 1: (a) 150 · π/180 = 15π/18 = 5π/6.
- Step 2: (b) −(π/4) · (180/π) = −45°.
- Step 3: (c) Subtract one full turn: 17π/6 − 12π/6 = 5π/6, which is between 0 and 2π.
Answer: (a) 5π/6, (b) −45°, (c) 5π/6.
- Example 2
Trig values from a point on the ray
The terminal ray of θ passes through (−3, 4). Find sin θ, cos θ and tan θ.
Show the solutionHide the solution
- Step 1: The distance from the origin to (−3, 4) is r = √((−3)² + 4²) = √25 = 5.
- Step 2: sin θ = y/r = 4/5 and cos θ = x/r = −3/5.
- Step 3: tan θ = y/x = 4/(−3) = −4/3. The point is in quadrant II, where sine is positive and cosine and tangent are negative, which matches.
Answer: sin θ = 4/5, cos θ = −3/5, tan θ = −4/3.
- Example 3
Trap: arc length needs radians
A circle has radius 10 cm. Find the length of the arc cut off by a central angle of 72°.
Show the solutionHide the solution
- Step 1: s = rθ works only with θ in radians. Writing 10 · 72 = 720 cm is the trap; that arc would be longer than the whole circle.
- Step 2: Convert: 72° = 72π/180 = 2π/5 radians.
- Step 3: s = 10 · 2π/5 = 4π ≈ 12.566 cm.
Answer: 4π cm, about 12.566 cm.
Common mistakes
- Using s = rθ with degrees. Convert to radians first.
- Mixing up sine and cosine on the unit circle. Cosine goes with x (horizontal), sine with y (vertical), in the same alphabetical order as (x, y).
- Forgetting signs by quadrant. In quadrant II, cos θ and tan θ are negative even though the triangle you draw has positive side lengths.
On the exam
- The exam is in radians. On calculator questions, check that your calculator is in radian mode before you start.
- Expect questions that give a point on a terminal ray or on a circle and ask for a trig value, or that ask which angle is coterminal with a given one.
Connected topics
Videos
Check yourself
4 questions on 3.2 Sine, Cosine, and Tangent. Pick an answer to see if you got it, and why.
An angle θ in standard position has a terminal ray that passes through the point (−3, 4). What is the value of tan θ?
In a circle of radius 6 centimeters centered at the origin, an angle in standard position cuts off an arc of length 15 centimeters. What is the radian measure of the angle?
Through how many radians does the minute hand of a clock turn in 20 minutes?
An angle θ in standard position has its terminal ray in Quadrant III, and tan θ = 3/4. What is the value of sin θ?
0 of 4 answered