Questions and Answers
What is the value of sin(60°)?
What is the relationship expressed in the Pythagorean identity?
Which of the following statements about cosine is true?
What is the tangent of 45°?
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Which ratio corresponds to the cosecant function?
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Study Notes
Trigonometric Ratios
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Definition: Trigonometric ratios relate the angles of a right triangle to the ratios of its sides.
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Basic Ratios:
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Sine (sin):
- sin(θ) = Opposite side / Hypotenuse
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Cosine (cos):
- cos(θ) = Adjacent side / Hypotenuse
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Tangent (tan):
- tan(θ) = Opposite side / Adjacent side
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Sine (sin):
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Reciprocal Ratios:
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Cosecant (csc):
- csc(θ) = 1/sin(θ) = Hypotenuse / Opposite side
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Secant (sec):
- sec(θ) = 1/cos(θ) = Hypotenuse / Adjacent side
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Cotangent (cot):
- cot(θ) = 1/tan(θ) = Adjacent side / Opposite side
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Cosecant (csc):
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Important Angles:
- For 0°, 30°, 45°, 60°, and 90°:
- sin(0°) = 0, sin(30°) = 1/2, sin(45°) = √2/2, sin(60°) = √3/2, sin(90°) = 1
- cos(0°) = 1, cos(30°) = √3/2, cos(45°) = √2/2, cos(60°) = 1/2, cos(90°) = 0
- tan(0°) = 0, tan(30°) = 1/√3, tan(45°) = 1, tan(60°) = √3, tan(90°) is undefined
- For 0°, 30°, 45°, 60°, and 90°:
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Identities:
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Pythagorean Identity:
- sin²(θ) + cos²(θ) = 1
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Sum and Difference Formulas:
- sin(A ± B) = sinA cosB ± cosA sinB
- cos(A ± B) = cosA cosB ∓ sinA sinB
- tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB)
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Pythagorean Identity:
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Applications:
- Used in solving triangles (finding sides and angles).
- Essential in physics, engineering, and various fields of science.
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Visualization:
- Right triangle representation helps in understanding the ratios.
- Unit circle can also be utilized for understanding angles beyond 0° to 360°.
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Calculator Use:
- Ensure calculator is in the correct mode (degrees or radians) based on the angle measurement needed.
Trigonometric Ratios Overview
- Trigonometric ratios are vital in connecting the angles of a right triangle to their side lengths.
Basic Ratios
- Sine (sin): Represents the ratio of the length of the opposite side to the hypotenuse.
- Cosine (cos): Represents the ratio of the length of the adjacent side to the hypotenuse.
- Tangent (tan): Represents the ratio of the length of the opposite side to the adjacent side.
Reciprocal Ratios
- Cosecant (csc): The reciprocal of sine; calculated as hypotenuse divided by the opposite side.
- Secant (sec): The reciprocal of cosine; calculated as hypotenuse divided by the adjacent side.
- Cotangent (cot): The reciprocal of tangent; calculated as adjacent side divided by the opposite side.
Important Angles
- Key values of sine, cosine, and tangent for standard angles:
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Sine:
- sin(0°) = 0
- sin(30°) = 1/2
- sin(45°) = √2/2
- sin(60°) = √3/2
- sin(90°) = 1
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Cosine:
- cos(0°) = 1
- cos(30°) = √3/2
- cos(45°) = √2/2
- cos(60°) = 1/2
- cos(90°) = 0
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Tangent:
- tan(0°) = 0
- tan(30°) = 1/√3
- tan(45°) = 1
- tan(60°) = √3
- tan(90°) is undefined.
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Sine:
Identities
- Pythagorean Identity: Fundamental identity of trigonometry, stating that the squares of sine and cosine of an angle sum to one: sin²(θ) + cos²(θ) = 1.
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Sum and Difference Formulas:
- sin(A ± B) = sinA cosB ± cosA sinB
- cos(A ± B) = cosA cosB ∓ sinA sinB
- tan(A ± B) = (tanA ± tanB) / (1 ∓ tanA tanB).
Applications
- Essential for resolving triangles to find unknown sides and angles.
- Widely used in fields such as physics and engineering, highlighting their importance across the sciences.
Visualization
- Right triangles provide a clear representation for understanding trigonometric ratios.
- The unit circle serves as a tool to grasp angles outside the range of 0° to 360°.
Calculator Use
- Ensure that the calculator is set to the appropriate mode (degrees or radians) to match the angle measurement required for calculations.
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Description
This quiz covers the fundamental trigonometric ratios, including sine, cosine, and tangent, as well as their reciprocal counterparts. It also explores important angles and their corresponding ratio values, providing a solid foundation for understanding how these concepts apply in right triangles.