Week 12 · Trigonometric functions: the unit circle, signs by quadrant and the ratios of special angles · 3 lessons
1
The unit circle: cos θ = x, sin θ = y, tan θ = y/x and the reciprocals sec θ, csc θ and cot θ
Mathematics Sec 1 Term 1, Unit 4, Lesson 3
GoalRead the six trigonometric ratios of an angle from a point on its terminal side.
New words
unit circle · the circle with centre O and radius 1sine and cosine · sin θ = y and cos θ = x for the point on the unit circletangent · tan θ = y/x
Explanation
Put the angle θ in standard position. Its terminal side meets the unit circle (centre O, radius 1) at a point P(x, y). Then cos θ = x, sin θ = y and tan θ = y/x (x ≠ 0). Their reciprocals are sec θ = 1/x (the secant), csc θ = 1/y (the cosecant) and cot θ = x/y (the cotangent). Because P is on the circle, x² + y² = 1, so sin²θ + cos²θ = 1. For any point (x, y) on the terminal side, not necessarily on the unit circle, let r = √(x² + y²) be its distance from O: sin θ = y/r, cos θ = x/r and tan θ = y/x. In the Arabic books the names are written with the letters جا, جتا and ظا; here we keep the Latin sin, cos and tan.
Ratio
On the unit circle
Point (x, y), r = √(x² + y²)
sin θ
y
y/r
cos θ
x
x/r
tan θ
y/x
y/x
csc θ
1/y
r/y
sec θ
1/x
r/x
cot θ
x/y
x/y
Worked examples
The terminal side of θ passes through P(−0.6, 0.8) on the unit circle. Then sin θ = 0.8, cos θ = −0.6 and tan θ = 0.8/(−0.6) = −4/3. Check: (−0.6)² + 0.8² = 0.36 + 0.64 = 1 ✓.
The terminal side passes through (−5, 12). r = √(25 + 144) = 13. sin θ = 12/13, cos θ = −5/13, tan θ = −12/5; csc θ = 13/12, sec θ = −13/5, cot θ = −5/12.
If cos θ = 0.5 then sec θ = 1/0.5 = 2. The reciprocal flips the number, and sec θ is never between −1 and 1.
Class activity
Shadow stick. Draw a circle of radius 10 cm on squared paper with centre at the origin. Mark the point on the circle at 60° with a protractor. Read its x and y from the grid and divide each by 10. Compare with cos 60° = 0.5 and sin 60° ≈ 0.87.
Practice
1
P(0.28, 0.96) is on the unit circle at angle θ. Find sin θ, cos θ and tan θ.
sin θ = 0.96, cos θ = 0.28, tan θ = 0.96/0.28 = 24/7.
2
The terminal side passes through (3, 4). Find the six ratios.
r = 5: sin 4/5, cos 3/5, tan 4/3, csc 5/4, sec 5/3, cot 3/4.
3
The terminal side passes through (−7, −24). Find sin θ and cos θ.
r = 25: sin θ = −24/25, cos θ = −7/25.
4
Is there an angle with sin θ = 1.2? Explain.
No. sin θ = y and a point on the unit circle has y between −1 and 1.
