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Questions and Answers
What is the Pythagorean identity used for in solving right triangles?
What is the domain of the inverse sine function?
What is the period of the sine function?
What is the formula for the law of cosines?
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What is the application of trigonometry in modeling sound waves and light waves?
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What is the formula for the double angle formula for sine?
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Study Notes
Solving Triangles
- Right Triangle Trigonometry:
- Use sine, cosine, and tangent to relate angle measures to side lengths
- Use Pythagorean identity: a^2 + b^2 = c^2 (c = hypotenuse)
- Non-Right Triangle Trigonometry:
- Use law of sines: a/sin(A) = b/sin(B) = c/sin(C)
- Use law of cosines: c^2 = a^2 + b^2 - 2ab * cos(C)
Inverse Trig Functions
- Inverse Sine (arcsin):
- Defined as: arcsin(x) = θ, where sin(θ) = x, -π/2 ≤ θ ≤ π/2
- Inverse Cosine (arccos):
- Defined as: arccos(x) = θ, where cos(θ) = x, 0 ≤ θ ≤ π
- Inverse Tangent (arctan):
- Defined as: arctan(x) = θ, where tan(θ) = x, -π/2 < θ < π/2
- Identities:
- arcsin(x) = arccos(√(1 - x^2))
- arctan(x) = arccos(1/√(1 + x^2))
Trigonometric Identities
- Pythagorean Identity:
- sin^2(θ) + cos^2(θ) = 1
- Sum and Difference Formulas:
- sin(a + b) = sin(a)cos(b) + cos(a)sin(b)
- cos(a + b) = cos(a)cos(b) - sin(a)sin(b)
- Double Angle Formulas:
- sin(2θ) = 2sin(θ)cos(θ)
- cos(2θ) = cos^2(θ) - sin^2(θ)
Graphing Trigonometric Functions
- Graph of Sine:
- Period: 2Ï€
- Amplitude: 1
- Graph of Cosine:
- Period: 2Ï€
- Amplitude: 1
- Graph of Tangent:
- Period: π
- Vertical asymptotes at π/2 + kπ (k ∈ ℤ)
Applications of Trigonometry
- Right Triangle Applications:
- Height and distance problems
- Angle of elevation and depression
- Wave Applications:
- Modeling sound waves and light waves
- Analyzing periodic phenomena
- Analytic Trigonometry:
- Finding coordinates of points on a unit circle
- Solving trigonometric equations using inverse functions
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Description
Test your understanding of trigonometry concepts, including right and non-right triangle trigonometry, inverse trig functions, identities, graphing, and applications. Evaluate your knowledge of sine, cosine, tangent, and more.
