Questions and Answers
If angle A and angle B are complementary, and sin(A) = 1/2, what is cos(B)?
Which of the following is NOT a trigonometric identity?
In a right triangle, the length of the hypotenuse is 10 and the length of one leg is 6. What is the sine of the angle opposite the leg of length 6?
What is the value of tan(45°)?
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What is the value of cos(120°)?
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If sin(A) = 3/5 and cos(A) = 4/5, what is the value of tan(A)?
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If sin(A) = 1/3, what is the value of cos(2A)?
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What is the value of sin(180° - A) in terms of sin(A)?
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Study Notes
Angles and Triangles
- Degrees and Radians: Angles can be measured in degrees or radians. 1 radian is equivalent to 180/π degrees.
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Angle Properties:
- Complementary Angles: Two angles that add up to 90° (π/2 radians).
- Supplementary Angles: Two angles that add up to 180° (π radians).
- Adjacent Angles: Two angles that share a common vertex and side.
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Triangles:
- Right Triangles: Triangles with one right angle (90°).
- Oblique Triangles: Triangles that are not right triangles.
Sine, Cosine, and Tangent
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Sine (sin): Ratio of the opposite side to the hypotenuse.
- sin(A) = opposite side / hypotenuse
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Cosine (cos): Ratio of the adjacent side to the hypotenuse.
- cos(A) = adjacent side / hypotenuse
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Tangent (tan): Ratio of the opposite side to the adjacent side.
- tan(A) = opposite side / adjacent side
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SOH-CAH-TOA: A mnemonic to remember the trig ratios:
- Sine = Opposite / Hypotenuse
- Cosine = Adjacent / Hypotenuse
- Tangent = Opposite / Adjacent
Trigonometric Identities
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Pythagorean Identity:
- sin²(A) + cos²(A) = 1
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Sum and Difference Identities:
- sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
- sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
- cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
- cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
-
Double Angle Identities:
- sin(2A) = 2sin(A)cos(A)
- cos(2A) = cos²(A) - sin²(A)
Angles and Triangles
- Angles can be measured in degrees or radians, with 1 radian equivalent to 180/π degrees.
- Complementary angles are two angles that add up to 90° (π/2 radians).
- Supplementary angles are two angles that add up to 180° (π radians).
- Adjacent angles are two angles that share a common vertex and side.
- Right triangles have one right angle (90°).
- Oblique triangles are triangles that are not right triangles.
Sine, Cosine, and Tangent
- Sine (sin) is the ratio of the opposite side to the hypotenuse, calculated as sin(A) = opposite side / hypotenuse.
- Cosine (cos) is the ratio of the adjacent side to the hypotenuse, calculated as cos(A) = adjacent side / hypotenuse.
- Tangent (tan) is the ratio of the opposite side to the adjacent side, calculated as tan(A) = opposite side / adjacent side.
- The SOH-CAH-TOA mnemonic helps remember the trig ratios: Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent.
Trigonometric Identities
- The Pythagorean Identity states that sin²(A) + cos²(A) = 1.
- The Sum and Difference Identities are:
- sin(A + B) = sin(A)cos(B) + cos(A)sin(B)
- sin(A - B) = sin(A)cos(B) - cos(A)sin(B)
- cos(A + B) = cos(A)cos(B) - sin(A)sin(B)
- cos(A - B) = cos(A)cos(B) + sin(A)sin(B)
- The Double Angle Identities are:
- sin(2A) = 2sin(A)cos(A)
- cos(2A) = cos²(A) - sin²(A)
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Description
Learn about angles and triangles, including degrees and radians, complementary, supplementary, and adjacent angles, and right and oblique triangles.
