Trig Ratios for Any Angle: CAST & Reference Angles (Grade 10)

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Grade 10 · Trigonometry

MEMORANDUM

Name:

Class:

Date:

Mark

/ 29

Answer all questions. Show all your working — marks are awarded for method as well as the final answer. Teacher copy: accept any correct equivalent method.

1
Write down the quadrant in which 235° lies, and state which of sin, cos and tan is positive there.
(2)
Quadrant 3; only tan is positive.✓✓ (2)
2
Determine the reference angle for θ = 295°.
(2)
Q4, so β = 360° − 295° = 65°✓✓ (2)
3
Evaluate cos 150° without using a calculator.
(3)
Q2, β = 30°, cos is negative in Q2, so cos 150° = −cos 30° = −(√3)/2✓✓✓ (3)
4
Evaluate tan 225° without using a calculator.
(3)
Q3, β = 45°, tan is positive in Q3, so tan 225° = +tan 45° = 1✓✓✓ (3)
5
P(−8 ; −6) lies on the arm of θ. Determine sinθ, cosθ and tanθ.
(4)
r = √(64 + 36) = 10 ✓; sinθ = −6/10 = −3/5 ✓; cosθ = −8/10 = −4/5 ✓; tanθ = −6/−8 = 3/4 ✓ (Q3, so only tan is positive)(4)
6
If cosθ = 8/17 and sinθ < 0, determine sinθ and tanθ.
(4)
cos positive, sin negative → Q4. x = 8, r = 17, y² = 289 − 64 = 225, y = ±15; in Q4 y is negative so y = −15. sinθ = −15/17, tanθ = −15/8✓✓✓✓ (4)
7
Write down the value of cosec 30°.
(2)
cosec 30° = 1 ÷ sin 30° = 1 ÷ ½ = 2✓✓ (2)
8
Explain why sec 90° is undefined.
(2)
secθ = 1/cosθ and cos 90° = 0; division by zero is undefined.✓✓ (2)
9
Evaluate cosec 210° without using a calculator.
(3)
210° is in Q3, so β = 210° − 180° = 30°; in Q3 only tan is positive, so sin 210° = −sin 30° = −½; cosec 210° = 1 ÷ (−½) = −2✓✓✓ (3)
10
Without using a calculator, show that sin 150° + cos 300° = 1.
(4)
sin 150°: Q2, β = 180° − 150° = 30°, and sin is positive in Q2, so sin 150° = +sin 30° = ½; cos 300°: Q4, β = 360° − 300° = 60°, and cos is positive in Q4, so cos 300° = +cos 60° = ½; ½ + ½ = 1✓✓✓✓ (4)

TOTAL: 29 marks