IB AA · Unit 2 · 18B Trigonometric Functions
Find the reference angle on a special triangle, then use the quadrant to choose the sign.
Special triangles
30°: opposite 1, adjacent , hypotenuse 2.
45°: opposite 1, adjacent 1, hypotenuse .
60°: opposite , adjacent 1, hypotenuse 2.
The sign
Quadrant I: sine, cosine, and tangent are positive.
Quadrant II: sine is positive.
Quadrant III: tangent is positive.
Quadrant IV: cosine is positive.
More than one turn
Subtract or add 2π, or 360°, until the angle is on the first turn.
Then use the reference angle.
Read the sides. Rationalize tangent of 30°.
The reference angle is 60°. The side opposite 60° is , the adjacent side is 1, and the hypotenuse is 2.
sin 60° = opphyp = 2
sin π3 = 2
Answer: 2
The reference angle is 30°. The side opposite 30° is 1, the adjacent side is , and the hypotenuse is 2.
cos 30° = adjhyp = 2
cos π6 = 2
Answer: 2
The reference angle is 45°. Both legs are 1, and the hypotenuse is .
tan 45° = oppadj = 1
tan π4 = 1
Answer: 1
The reference angle is 30°. The side opposite 30° is 1, the adjacent side is , and the hypotenuse is 2.
sin 30° = opphyp = 12
sin 30° = 12
Answer: 12
The reference angle is 60°. The side opposite 60° is , the adjacent side is 1, and the hypotenuse is 2.
cos 60° = adjhyp = 12
cos 60° = 12
Answer: 12
The reference angle is 30°. The side opposite 30° is 1, the adjacent side is , and the hypotenuse is 2.
tan 30° = oppadj = 1
Multiply the top and bottom by .
tan 30° = 3
Answer: 3
The absolute value comes from the reference angle. The quadrant chooses the sign.
The reference angle is 45°. Both legs are 1, and the hypotenuse is .
sin 45° = opphyp = 2
Sin is negative in Quadrant III.
sin 5π4 = −2
Answer: −2
The reference angle is 60°. The side opposite 60° is , the adjacent side is 1, and the hypotenuse is 2.
cos 60° = adjhyp = 12
Cos is negative in Quadrant II.
cos 2π3 = −12
Answer: −12
The reference angle is 30°. The side opposite 30° is 1, the adjacent side is , and the hypotenuse is 2.
tan 30° = oppadj = 1
Multiply the top and bottom by .
tan 30° = 3
Tan is positive in Quadrant III.
tan 7π6 = 3
Answer: 3
The reference angle is 60°. The side opposite 60° is , the adjacent side is 1, and the hypotenuse is 2.
sin 60° = opphyp = 2
Sin is negative in Quadrant III.
sin 240° = −2
Answer: −2
The reference angle is 30°. The side opposite 30° is 1, the adjacent side is , and the hypotenuse is 2.
cos 30° = adjhyp = 2
Cos is positive in Quadrant IV.
cos 330° = 2
Answer: 2
The reference angle is 30°. The side opposite 30° is 1, the adjacent side is , and the hypotenuse is 2.
tan 30° = oppadj = 1
Multiply the top and bottom by .
tan 30° = 3
Tan is negative in Quadrant IV.
tan 11π6 = −3
Answer: −3
Land on one turn before you name the quadrant.
11π3 − 2π = 5π3
The reference angle is 60°. The side opposite 60° is , the adjacent side is 1, and the hypotenuse is 2.
sin 60° = opphyp = 2
Sin is negative in Quadrant IV.
sin 11π3 = −2
Answer: −2
−5π4 + 2π = 3π4
The reference angle is 45°. Both legs are 1, and the hypotenuse is .
cos 45° = adjhyp = 2
Cos is negative in Quadrant II.
cos(−5π4) = −2
Answer: −2
−2π3 + 2π = 4π3
The reference angle is 60°. The side opposite 60° is , the adjacent side is 1, and the hypotenuse is 2.
tan 60° = oppadj =
Tan is positive in Quadrant III.
tan(−2π3) =
Answer:
510° − 360° = 150°
The reference angle is 30°. The side opposite 30° is 1, the adjacent side is , and the hypotenuse is 2.
sin 30° = opphyp = 12
Sin is positive in Quadrant II.
sin 510° = 12
Answer: 12
−7π6 + 2π = 5π6
The reference angle is 30°. The side opposite 30° is 1, the adjacent side is , and the hypotenuse is 2.
cos 30° = adjhyp = 2
Cos is negative in Quadrant II.
cos(−7π6) = −2
Answer: −2
660° − 360° = 300°
The reference angle is 60°. The side opposite 60° is , the adjacent side is 1, and the hypotenuse is 2.
tan 60° = oppadj =
Tan is negative in Quadrant IV.
tan 660° = −
Answer: −
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