5 Questions
What is the formula for sin(2θ) in terms of θ?
2sin(θ)cos(θ)
What is the formula for cos(2θ) in terms of θ?
cos²(θ) - sin²(θ)
What is the formula for tan(θ/2) in terms of θ?
(1 - cos(θ)) / sin(θ)
What is the formula for sin(θ/2) in terms of θ?
±√((1 - cos(θ))/2)
What is the formula for cos(θ/2) in terms of θ?
±√((1 + cos(θ))/2)
Study Notes
Trigonometric Identities
Double Angle Formulas
-
Sine Double Angle Formula:
- sin(2θ) = 2sin(θ)cos(θ)
-
Cosine Double Angle Formulas:
- cos(2θ) = cos²(θ) - sin²(θ) = 1 - 2sin²(θ) = 2cos²(θ) - 1
-
Tangent Double Angle Formula:
- tan(2θ) = 2tan(θ) / (1 - tan²(θ))
Half Angle Formulas
-
Sine Half Angle Formula:
- sin(θ/2) = ±√((1 - cos(θ))/2)
-
Cosine Half Angle Formula:
- cos(θ/2) = ±√((1 + cos(θ))/2)
-
Tangent Half Angle Formula:
- tan(θ/2) = ±√((1 - cos(θ))/(1 + cos(θ)))
- tan(θ/2) = sin(θ) / (1 + cos(θ))
- tan(θ/2) = (1 - cos(θ)) / sin(θ)
Trigonometric Identities
Double Angle Formulas
- Double angle formulas are used to find the sine, cosine, and tangent of twice an angle.
- Sine Double Angle Formula: sin(2θ) = 2sin(θ)cos(θ)
-
Cosine Double Angle Formulas: cos(2θ) has three forms:
- cos²(θ) - sin²(θ)
- 1 - 2sin²(θ)
- 2cos²(θ) - 1
- Tangent Double Angle Formula: tan(2θ) = 2tan(θ) / (1 - tan²(θ))
Half Angle Formulas
- Half angle formulas are used to find the sine, cosine, and tangent of half an angle.
- Sine Half Angle Formula: sin(θ/2) = ±√((1 - cos(θ))/2)
- Cosine Half Angle Formula: cos(θ/2) = ±√((1 + cos(θ))/2)
-
Tangent Half Angle Formula: tan(θ/2) has three forms:
- ±√((1 - cos(θ))/(1 + cos(θ)))
- sin(θ) / (1 + cos(θ))
- (1 - cos(θ)) / sin(θ)
Test your knowledge of trigonometric identities, including double angle formulas and half angle formulas for sine, cosine, and tangent.
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