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Questions and Answers
What is the Pythagorean Identity, and how is it used to verify trigonometric identities?
What is the Pythagorean Identity, and how is it used to verify trigonometric identities?
The Pythagorean Identity is sin²θ + cos²θ = 1. It is used to verify trigonometric identities by substituting sin²θ or cos²θ into the identity to simplify expressions.
Derive the sum formula for cosine, cos(a + b) = cos(a)cos(b) - sin(a)sin(b), using the unit circle.
Derive the sum formula for cosine, cos(a + b) = cos(a)cos(b) - sin(a)sin(b), using the unit circle.
Starting with the unit circle, we can derive the formula by using the x- and y-coordinates of points on the circle.
Prove that tan(2θ) = (2tan(θ)) / (1 - tan²(θ)) using the double angle formula for sine and cosine.
Prove that tan(2θ) = (2tan(θ)) / (1 - tan²(θ)) using the double angle formula for sine and cosine.
Start with the double angle formulas for sine and cosine, then divide the sine formula by the cosine formula to obtain the tangent formula.
What is the cotangent of an angle, and how is it related to the sine and cosine ratios?
What is the cotangent of an angle, and how is it related to the sine and cosine ratios?
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Solve the right triangle with legs 3 and 4, and find the length of the hypotenuse using the Pythagorean theorem.
Solve the right triangle with legs 3 and 4, and find the length of the hypotenuse using the Pythagorean theorem.
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In an oblique triangle, if a = 5, b = 7, and C = 60°, find the length of side c using the law of cosines.
In an oblique triangle, if a = 5, b = 7, and C = 60°, find the length of side c using the law of cosines.
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How is trigonometry used in medical imaging to reconstruct 3D models from 2D images?
How is trigonometry used in medical imaging to reconstruct 3D models from 2D images?
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What is the law of sines, and how is it used to solve triangles?
What is the law of sines, and how is it used to solve triangles?
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Derive the double angle formula for sine, sin(2θ) = 2sin(θ)cos(θ), using the sum formula for sine.
Derive the double angle formula for sine, sin(2θ) = 2sin(θ)cos(θ), using the sum formula for sine.
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How is trigonometry used in navigation to determine distances and directions between locations?
How is trigonometry used in navigation to determine distances and directions between locations?
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Study Notes
Identities and Formulas
- Pythagorean Identity: sin²θ + cos²θ = 1
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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)
- tan(a + b) = (tan(a) + tan(b)) / (1 - tan(a)tan(b))
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Double and Half Angle Formulas:
- sin(2θ) = 2sin(θ)cos(θ)
- cos(2θ) = cos²(θ) - sin²(θ)
- tan(2θ) = (2tan(θ)) / (1 - tan²(θ))
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Product-to-Sum Formulas:
- sin(a)cos(b) = (1/2)[sin(a + b) + sin(a - b)]
- cos(a)sin(b) = (1/2)[sin(a + b) - sin(a - b)]
Trigonometric Ratios
- Sine (sin): opposite side / hypotenuse
- Cosine (cos): adjacent side / hypotenuse
- Tangent (tan): opposite side / adjacent side
- Cotangent (cot): adjacent side / opposite side
- Secant (sec): hypotenuse / adjacent side
- Cosecant (csc): hypotenuse / opposite side
Solving Triangles
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Right Triangles:
- Use trigonometric ratios to find missing sides or angles
- Apply Pythagorean theorem to find hypotenuse
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Oblique Triangles:
- Use law of sines or law of cosines to find missing sides or angles
- Law of sines: a / sin(A) = b / sin(B) = c / sin(C)
- Law of cosines: c² = a² + b² - 2ab * cos(C)
Applications of Trigonometry
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Geometry and Physics:
- Calculate distances, heights, and angles in triangles
- Model periodic phenomena (e.g., sound waves, light waves)
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Navigation and Surveying:
- Determine distances and directions between locations
- Calculate heights and angles of landmarks
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Medical Imaging and Computer Graphics:
- Use trigonometry to reconstruct 3D models from 2D images
- Apply trigonometric transformations to manipulate objects in 3D space
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Description
Test your knowledge of trigonometric identities, formulas, and applications in geometry, physics, navigation, and more. Covers Pythagorean identity, sum and difference formulas, and solving triangles.
