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- Trigonometric ratios of any angle
Trigonometric ratios of any angle
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sin, cos and tan work for any angle, not just those in a right-angled triangle. Exact values and symmetry find them without a calculator.
Exact values
Learn these:
Signs in each quadrant
0° to 90°: all positive
90° to 180°: only sin positive
180° to 270°: only tan positive
270° to 360°: only cos positive
90° to 180°: only sin positive
180° to 270°: only tan positive
270° to 360°: only cos positive
Related angles
Find the acute angle to the x-axis, then fix the sign.
sin 120° = sin 60° = √32 (sin is positive)
cos 300° = cos 60° = 12 (cos is positive)
tan 225° = tan 45° = 1 (tan is positive)
sin 120° = sin 60° = √32 (sin is positive)
cos 300° = cos 60° = 12 (cos is positive)
tan 225° = tan 45° = 1 (tan is positive)
Find the exact value of cos 150°.
- The acute angle to the x-axis is 30°
- 150° is in the second quadrant, where cos is negative
- cos 30° = √32
Answer: cos 150° = −√32
Use the acute angle to the x-axis for the size and the quadrant for the sign. Know the exact values for 30°, 45° and 60°.
The interactive lesson includes the diagrams for this topic.
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