Trigonometric functions start from a point moving round the unit circle, the circle of radius 1. After turning through an angle θ, the point's x-coordinate is the cosine, its y-coordinate is the sine, and y divided by x is the tangent.
That lets trigonometric ratios, once limited to the acute angles of a right triangle, be defined for any angle, with signs decided by the quadrant the point lands in.
The single fact that the radius is 1 also gives the relation sin²θ + cos²θ = 1 straight away.
In the first picture, changing the angle sends the point round the circle and changes the lengths of the red line (cos) and the blue line (sin); the second gathers the positions of the special angles.
A point on the unit circle: defining sin and cos
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The point reached by turning through θ on the unit circle, with its x-coordinate cos θ (red) and y-coordinate sin θ (blue)
45°
💡 The unit circle is where trigonometric functions begin
①The point reached by turning through θ on a circle of radius 1 is P(cos θ, sin θ)
②cos θ is the x-coordinate (red line) and sin θ is the y-coordinate (blue line)
③In quadrants where a coordinate is negative, cos θ or sin θ is negative too
④At θ = 90° and 270°, cos θ = 0, so tan θ is undefined
Values at the special angles
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The positions of the special angles (multiples of 30° and 45°) on the unit circle; the red points are 0°, 90°, 180° and 270° on the axes
Ratios for 30° and 60°
sin 30° = 12, cos 30° = √32, tan 30° = 1√3
A right triangle with angles 30°, 60°, 90° has sides in the ratio 1 : √3 : 2
Ratios for 45°
sin 45° = √22, cos 45° = √22, tan 45° = 1
Half of a square cut along its diagonal, with sides in the ratio 1 : 1 : √2
Basic relations between the functions
Pythagorean identity
sin²θ + cos²θ = 1
P(cos θ, sin θ) lies on the circle of radius 1, so x² + y² = 1
Definition of tangent
tan θ = sin θcos θ
It equals the slope y/x of the radius from the origin to P
Reciprocal functions
sec θ = 1cos θ, csc θ = 1sin θ, cot θ = 1tan θ
The reciprocals of the three functions have names of their own
Signs by quadrant
📐 Quadrants and signs
①First quadrant (0° < θ < 90°): sin, cos and tan all positive
②Second quadrant (90° < θ < 180°): only sin positive
③Third quadrant (180° < θ < 270°): only tan positive
④Fourth quadrant (270° < θ < 360°): only cos positive
⑤In order, "All, Sin, Tan, Cos", often remembered as ASTC
Summary
The heart of trigonometric functions
sin²θ + cos²θ = 1, tan θ = sin θcos θ
Every trigonometric function is defined by a single point on the unit circle
🎯 Exam Points
①On the unit circle, P(cos θ, sin θ)
②Ratios for the special angles 30°, 45°, 60°
③sin²θ + cos²θ = 1
④Signs by quadrant: all, sin, tan, cos positive in turn
⑤tan θ = sin θ / cos θ is undefined when cos θ = 0
Worked Examples & Past Exam
Example 1
If sin θ = 3/5 and θ is in the first quadrant, find cos θ.
1
Find cos²θ from sin²θ + cos²θ = 1.
cos²θ = 1 - sin²θ = 1 - (3/5)² = 16/25
2
In the first quadrant cos θ > 0, so take the positive square root.
cos θ = √(16/25) = 4/5
▸ cos θ = 4/5
Picture a right triangle with hypotenuse 5 and height 3; its base is 4, so cos θ = 4/5. The identity gives the size and the quadrant gives the sign.
Example 2
If tan θ = 1 and θ is in the third quadrant, find sin θ.
1
tan θ = 1, so the reference angle is 45°, and the third-quadrant angle is 225°.
θ = 180° + 45° = 225°
2
In the third quadrant sin θ < 0, so sin 225° = −sin 45°.
sin θ = -√22
▸ sin θ = −√2/2
A point in the third quadrant has both coordinates negative, so their ratio tan θ is positive. That is why tan θ = 1 is possible in the third quadrant too.
2023 CSAT Math type, adapted
If cos θ = −1/2 and 0° < θ < 180°, find θ.
①60°
②120°
③135°
④150°
⑤240°
▸ ② 120°
1
cos 60° = 1/2, so the reference angle is 60°.
cos 60° = 12
2
For 0° < θ < 180°, cos θ < 0 means the second quadrant, so θ = 180° − 60°.