Podcast
Questions and Answers
What is the formula for tan(A + B)?
What is the formula for tan(A + B)?
What is the reference angle for an angle in standard position?
What is the reference angle for an angle in standard position?
What is the formula for cos(2A)?
What is the formula for cos(2A)?
What is the sine of an angle A?
What is the sine of an angle A?
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What is the Pythagorean identity?
What is the Pythagorean identity?
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What is the formula for sin(A + B)?
What is the formula for sin(A + B)?
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Study Notes
Trigonometric Identities
- Pythagorean Identity: sin²(A) + cos²(A) = 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)
- tan(A + B) = (tan(A) + tan(B)) / (1 - tan(A)tan(B))
-
Double Angle Formulas:
- sin(2A) = 2sin(A)cos(A)
- cos(2A) = cos²(A) - sin²(A)
- tan(2A) = (2tan(A)) / (1 - tan²(A))
Triangles And Angles
- Right Triangle: one angle is 90° (π/2 radians)
-
Angles in Standard Position:
- Initial side: x-axis (positive)
- Terminal side: where the angle ends (can be any quadrant)
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Reference Angle:
- The acute angle between the terminal side and the x-axis
- Used to find trig values for non-acute angles
Sine Cosine And Tangent
-
Sine (sin):
- Opposite side / Hypotenuse
- sin(A) = opposite side / hypotenuse
-
Cosine (cos):
- Adjacent side / Hypotenuse
- cos(A) = adjacent side / hypotenuse
-
Tangent (tan):
- Opposite side / Adjacent side
- tan(A) = opposite side / adjacent side
Trigonometric Identities
- The Pythagorean Identity is a fundamental concept in trigonometry, stating that sin²(A) + cos²(A) = 1.
- The Sum and Difference Formulas describe the trigonometric relationships between angles A and B:
- 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))
- The Double Angle Formulas provide trigonometric values for double angles:
- sin(2A) = 2sin(A)cos(A)
- cos(2A) = cos²(A) - sin²(A)
- tan(2A) = (2tan(A)) / (1 - tan²(A))
Triangles and Angles
- A right triangle has one angle measuring 90° (π/2 radians).
- Angles in standard position have their initial side on the x-axis (positive) and their terminal side in any quadrant.
- The reference angle is the acute angle between the terminal side and the x-axis, used to find trigonometric values for non-acute angles.
Sine, Cosine, and Tangent
- Sine (sin) is defined as the opposite side divided by the hypotenuse: sin(A) = opposite side / hypotenuse.
- Cosine (cos) is defined as the adjacent side divided by the hypotenuse: cos(A) = adjacent side / hypotenuse.
- Tangent (tan) is defined as the opposite side divided by the adjacent side: tan(A) = opposite side / adjacent side.
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Description
Test your understanding of trigonometric identities, including the Pythagorean identity, sum and difference formulas, and double angle formulas, as well as topics related to triangles and angles.
