D. Trigonometric Functions · Trigonometric Ratios, Identities and LawsAngles in Standard Position and the CAST Rule
D1.2About 70 minutes
Learning goals- I can draw an angle from 0° to 360° in standard position and name its quadrant.
- I can find sin θ, cos θ and tan θ from any point P(x, y) on the terminal arm.
- I can use the CAST rule to predict the sign of a trigonometric ratio.
- I can find all three primary ratios when I know one ratio and the quadrant.
Words to know
standard position
An angle drawn with its vertex at the origin and its initial arm on the positive x-axis. Positive angles turn counterclockwise.
initial arm
The fixed arm of an angle in standard position; it lies on the positive x-axis.
terminal arm
The arm that rotates to show the size of the angle.
quadrantal angle
An angle whose terminal arm lies on an axis: 0°, 90°, 180°, 270° or 360°.
CAST rule
A memory aid for the signs of the ratios: All positive in quadrant I, Sine in II, Tangent in III, Cosine in IV.
Angles bigger than 90°
A Ferris wheel at a fall fair, the hands of a clock and a turning car tire all rotate through angles much bigger than 90°. A right triangle cannot hold an angle of 140°, so we need a new way to define sine, cosine and tangent. The trick is to put the angle on a coordinate grid.
Standard position
An angle θ is in standard position when:
- its vertex is at the origin (0, 0),
- its initial arm lies along the positive x-axis, and
- it opens counterclockwise to its terminal arm.
The axes split the plane into four quadrants, numbered I, II, III and IV counterclockwise starting from the top right. An angle of 140° has its terminal arm in quadrant II; an angle of 250° is in quadrant III.
Ratios from a point
Pick any point P(x, y) on the terminal arm, other than the origin. Let r be its distance from the origin: r = √(x² + y²). Since r is a distance, r is always positive. Then:
- sin θ = y/r
- cos θ = x/r
- tan θ = y/x (when x ≠ 0)
For an angle in quadrant I these are exactly SOH CAH TOA: y is the opposite side, x is the adjacent side and r is the hypotenuse. The new part is that x and y can be negative, so the ratios can be negative too.
Worked exampleExample 1: A point in quadrant II
The point P(−3, 4) is on the terminal arm of an angle θ in standard position. Find the three primary ratios and the angle θ to the nearest tenth of a degree.
- r = √((−3)² + 4²) = √25 = 5.
- sin θ = y/r = 4/5, cos θ = x/r = −3/5, tan θ = y/x = 4/(−3) = −4/3.
- The acute angle between the terminal arm and the x-axis is tan⁻¹(4/3) ≈ 53.1°.
- The arm is in quadrant II, so θ = 180° − 53.1° = 126.9°. (Check: cos 126.9° ≈ −0.600 ✓)
sin θ = 4/5, cos θ = −3/5, tan θ = −4/3, θ ≈ 126.9°
Which ratios are positive? The CAST rule
r is always positive, so the signs depend only on x and y:
- Quadrant I: x > 0, y > 0. All three ratios are positive.
- Quadrant II: x < 0, y > 0. Only sine (y/r) is positive.
- Quadrant III: x < 0, y < 0. Only tangent (y/x, negative ÷ negative) is positive.
- Quadrant IV: x > 0, y < 0. Only cosine (x/r) is positive.
Reading the quadrants IV, I, II, III gives the letters C A S T.
Worked exampleExample 2: A point in quadrant III
The point (−5, −12) is on the terminal arm of θ. Find sin θ, cos θ and tan θ.
- r = √(25 + 144) = √169 = 13.
- sin θ = −12/13, cos θ = −5/13, tan θ = (−12)/(−5) = 12/5.
- Check with CAST: quadrant III, so only tangent is positive. ✓
sin θ = −12/13, cos θ = −5/13, tan θ = 12/5
Worked exampleExample 3: One ratio and the quadrant
tan θ = −2/3 and θ is in quadrant IV. Find the exact values of sin θ and cos θ.
- In quadrant IV, x > 0 and y < 0. tan θ = y/x = −2/3, so choose x = 3 and y = −2.
- r = √(9 + 4) = √13.
- sin θ = y/r = −2/√13 = −2√13/13.
- cos θ = x/r = 3/√13 = 3√13/13.
sin θ = −2√13/13, cos θ = 3√13/13
Quadrantal angles
When the terminal arm lies on an axis, choose the point at distance 1 from the origin. For 90° that point is (0, 1), so sin 90° = 1/1 = 1, cos 90° = 0/1 = 0, and tan 90° = 1/0 is undefined. This agrees with what you found in lesson 1.
| θ | 0° | 90° | 180° | 270° | 360° |
|---|
| point | (1, 0) | (0, 1) | (−1, 0) | (0, −1) | (1, 0) |
| sin θ | 0 | 1 | 0 | −1 | 0 |
| cos θ | 1 | 0 | −1 | 0 | 1 |
| tan θ | 0 | undefined | 0 | undefined | 0 |
Just for funSome students remember CAST as "All Students Take Calculus", reading counterclockwise from quadrant I. In Ontario that is not even a joke: Calculus and Vectors (MCV4U) is a Grade 12 course.
Hands-on activityMeasure your own coordinates
You need: graph paper (or a sheet with a grid you draw), a ruler, a protractor, a pencil, a calculator, and a piece of string about 15 cm long.
- Draw x- and y-axes crossing near the centre of the sheet.
- Tie the string around the pencil, hold the other end at the origin with a finger, and draw a circle with radius 10 cm. (A compass works too.)
- With the protractor, mark angles of 140°, 220° and 310° in standard position. Draw each terminal arm to the circle.
- Read the x- and y-coordinates where each arm meets the circle (in cm). Divide by 10 to get x/r and y/r.
- Compare with your calculator (degree mode): cos 140°, sin 140°, and so on.
What to look for: for 140° you should read about (−7.7, 6.4), giving cos 140° ≈ −0.77 and sin 140° ≈ 0.64. For 220°: about (−7.7, −6.4). For 310°: about (6.4, −7.7). The signs should match CAST.
Your turn
1The point (8, −6) is on the terminal arm of θ. Find sin θ, cos θ and tan θ.
r = 10. sin θ = −3/5, cos θ = 4/5, tan θ = −3/4.
2The point (−1, √3) is on the terminal arm of θ. Find the three primary ratios.
r = √(1 + 3) = 2. sin θ = √3/2, cos θ = −1/2, tan θ = √3/(−1) = −√3.
3In which quadrant is sin θ < 0 and tan θ > 0?
Quadrant III (only tangent is positive there).
4sin θ = 5/13 and θ is in quadrant II. Find cos θ and tan θ.
y = 5, r = 13, so x = −√(169 − 25) = −12 (negative in quadrant II). cos θ = −12/13, tan θ = −5/12.
5Give the exact values of cos 180°, sin 270° and tan 360°.
6Without a calculator, is cos 250° positive or negative?
250° is in quadrant III, where cosine is negative.
More practice: warm-up
1In which quadrant is the terminal arm of 200°?
2In which quadrant is the terminal arm of 310°?
3Find r for the point (−6, 8).
4Is tan θ positive or negative in quadrant II?
More practice: on your own
1The point (−2, −5) is on the terminal arm of θ. Find the exact primary ratios.
r = √29. sin θ = −5/√29 = −5√29/29, cos θ = −2√29/29, tan θ = 5/2 (positive in quadrant III ✓).
2cos θ = −4/5 and 180° < θ < 270°. Find sin θ and tan θ.
Quadrant III: x = −4, r = 5, y = −3. sin θ = −3/5, tan θ = (−3)/(−4) = 3/4.
3tan θ = −1 and θ is in quadrant II. Find sin θ, cos θ and θ.
Take x = −1, y = 1, r = √2. sin θ = √2/2, cos θ = −√2/2, θ = 180° − 45° = 135°.
4Explain why r is always positive, but x and y can be negative.
r is a distance from the origin, and distances are never negative. x and y are coordinates, which are negative to the left of the y-axis and below the x-axis.
5Explain why tan 90° is undefined.
The point (0, 1) is on the terminal arm, so tan 90° = y/x = 1/0. Division by zero is undefined.
Challenge
1The point (k, 12) is on the terminal arm of θ, r = 13 and θ is in quadrant II. Find k and cos θ.
k² + 144 = 169, so k = ±5. In quadrant II, x < 0, so k = −5 and cos θ = −5/13.
2sin θ = −√3/2 and tan θ > 0. Find cos θ, tan θ and θ (0° ≤ θ ≤ 360°).
sin < 0 and tan > 0 means quadrant III. Take y = −√3, r = 2, x = −1. cos θ = −1/2, tan θ = √3, θ = 180° + 60° = 240°.
3Use x, y and r to show that (sin θ)² + (cos θ)² = 1 for every angle θ.
(y/r)² + (x/r)² = (x² + y²)/r². Since r² = x² + y², the fraction equals 1. (You will use this identity a lot in lesson 6.)
Quiz
1. In standard position, the initial arm of an angle lies on the:
2. The point (−3, 4) is on the terminal arm of θ. What is cos θ?
3. In quadrant III, which ratio is positive?
5. What is r for the point (5, −12)?
6. θ is in quadrant IV. Which statement is true?
Big idea (D1.2): angles up to 360° are drawn in standard position on a grid, and the ratios are defined from any point on the terminal arm: sin θ = y/r, cos θ = x/r, tan θ = y/x. The signs follow the CAST rule.
How to help: Name an angle (say 215°) and ask your teen to point to the quadrant, then say which ratios are positive. Ask for a quick sketch every time; a sketch catches most sign errors.
Common difficulties:
- Giving r a negative sign. r is a distance and is always positive.
- Forgetting which coordinate goes with sine (y) and which with cosine (x).
- Numbering quadrants clockwise. They go counterclockwise, I to IV, starting top right.
- Using the calculator's angle directly: tan⁻¹(4/(−3)) gives −53.1°, not 126.9°. The sketch tells you which quadrant you need.