Trigonometric Identities & Equations — Practice
One identity comes straight from Pythagoras, and it unlocks nearly every trig equation.
What these questions cover
These questions use 10 ideas from the course. Follow any of them for a definition, where it is taught, and what it leads to.
- Reciprocal — The flip of a fraction. The reciprocal of 35 is 53.
- Isolate — Get the variable alone on one side.
- Identity — An equation true for every value, like 2(x + 3) = 2x + 6.
- Sine — sin θ = oppositehypotenuse.
- Cosine — cos θ = adjacenthypotenuse.
- Secant — A line that cuts the circle at two points.
- Reference angle — The acute angle between the terminal side and the x-axis.
- Pythagorean identity — sin2θ + cos2θ = 1.
- y-coordinate — The second number — how far up or down.
- Quadrant — One of the four regions the axes divide the plane into.
Problem 1
If sinθ=0.6, find sin2θ+cos2θ.
Worked answer
1
The Pythagorean identity is always 1
Problem 2
If cosθ=35 and θ is acute, find sinθ.
Worked answer
4/5
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Problem 3
If sinθ=12 and cosθ=√32, find tanθ as a fraction with a root.
Worked answer
1/sqrt(3)
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Problem 4
If sinθ=14, find cscθ.
Worked answer
4
11/4=4
Problem 5
If cosθ=12, find secθ.
Worked answer
2
11/2=2
Problem 6
If sinθ=35 and θ is in quadrant II, find cosθ.
Worked answer
-4/5
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Problem 7
Solve sinθ=12 for 0° ≤ θ < 360°. Give both answers, smallest first.
Worked answer
30,150
Reference angle 30°
30° and 180-30=150°
Problem 8
Solve cosθ=12 for 0° ≤ θ < 360°. Give both, smallest first.
Worked answer
60,300
Reference angle 60°
60° and 360-60=300°
Problem 9
Solve cosθ=-1 for 0° ≤ θ < 360°.
Worked answer
180
Only one angle gives -1
Problem 10
How many solutions does sinθ=0.4 have for 0° ≤ θ < 360°?
Worked answer
2
One in quadrant I, one in quadrant II
Problem 11
Simplify 1-cos2θ using an identity. Answer in the form sin^2.
Worked answer
sin^2
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Problem 12
Solve 2sinθ-1=0 for 0° ≤ θ < 360°. Give both, smallest first.
Worked answer
30,150
sinθ=12
30° and 150°
Problem 13
Solve sinθ=0 for 0° ≤ θ < 360°. Give both, smallest first.
Worked answer
0,180
At 0° and 180°
Problem 14
Solve 2sin2θ-sinθ-1=0. One solution is sinθ=1. What is the other value of sinθ?
Worked answer
-1/2
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Problem 15
What does sin2 30° equal?
- 0.25
- 0.5
- 1
- sin 900°
Worked answer
0.25
sin 30°=0.5
0.52=0.25
Problem 16
If tanθ=sinθcosθ and cosθ=0.6, sinθ=0.8, find tanθ.
Worked answer
4/3
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