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Questions and Answers
What is the length of the hypotenuse in a 30-60-90 triangle if the shorter leg is 5 units?
What is the length of the hypotenuse in a 30-60-90 triangle if the shorter leg is 5 units?
What is the reference angle for 150°?
What is the reference angle for 150°?
What is the value of cos(60°) in a unit circle?
What is the value of cos(60°) in a unit circle?
What is the value of tan(45°) in a unit circle?
What is the value of tan(45°) in a unit circle?
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What is the length of the longer leg in a 30-60-90 triangle if the shorter leg is 3 units?
What is the length of the longer leg in a 30-60-90 triangle if the shorter leg is 3 units?
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What is the value of sin(0°) in a unit circle?
What is the value of sin(0°) in a unit circle?
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What is the range of the sine function?
What is the range of the sine function?
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What is the period of the tangent function?
What is the period of the tangent function?
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What is the identity for sin(A) in terms of cosine?
What is the identity for sin(A) in terms of cosine?
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What is the formula for cos(A) in terms of sine and tangent?
What is the formula for cos(A) in terms of sine and tangent?
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What is the relationship between the sine and cosine of an angle?
What is the relationship between the sine and cosine of an angle?
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What is the identity for tan(A) in terms of sine and cosine?
What is the identity for tan(A) in terms of sine and cosine?
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Study Notes
Trigonometric Functions of Special Angles
Special Right Triangles
- 30-60-90 Triangles:
- Hypotenuse: 2 times the length of the shorter leg
- Longer leg: √3 times the length of the shorter leg
- Trigonometric values:
- sin(30°) = 1/2
- cos(30°) = √3/2
- tan(30°) = 1/√3
- 45-45-90 Triangles:
- Legs are equal in length
- Hypotenuse: √2 times the length of a leg
- Trigonometric values:
- sin(45°) = 1/√2
- cos(45°) = 1/√2
- tan(45°) = 1
Reference Angles
- An angle in the unit circle that has the same trigonometric values as the original angle
- Found by subtracting the angle from the closest multiple of 360° or adding the angle to the closest multiple of 360° *Used to simplify trigonometric expressions and find exact values
Unit Circle
- A circle with a radius of 1 unit, used to define trigonometric functions
- Centered at the origin (0, 0) of a coordinate plane
- Angles measured counterclockwise from the positive x-axis
- Used to visualize and calculate trigonometric functions
Exact Values
- sin(0°) = 0
- cos(0°) = 1
- tan(0°) = 0
- sin(30°) = 1/2
- cos(30°) = √3/2
- tan(30°) = 1/√3
- sin(45°) = 1/√2
- cos(45°) = 1/√2
- tan(45°) = 1
- sin(60°) = √3/2
- cos(60°) = 1/2
- tan(60°) = √3
Trigonometric Functions of Special Angles
Special Right Triangles
- 30-60-90 Triangles: Hypotenuse is 2 times the length of the shorter leg, and the longer leg is √3 times the length of the shorter leg.
- 30-60-90 Triangles: Trigonometric values are sin(30°) = 1/2, cos(30°) = √3/2, and tan(30°) = 1/√3.
- 45-45-90 Triangles: Legs are equal in length, and the hypotenuse is √2 times the length of a leg.
- 45-45-90 Triangles: Trigonometric values are sin(45°) = 1/√2, cos(45°) = 1/√2, and tan(45°) = 1.
Reference Angles
- A reference angle is an angle in the unit circle with the same trigonometric values as the original angle.
- To find a reference angle, subtract the angle from the closest multiple of 360° or add the angle to the closest multiple of 360°.
- Reference angles are used to simplify trigonometric expressions and find exact values.
Unit Circle
- The unit circle is a circle with a radius of 1 unit, used to define trigonometric functions.
- The unit circle is centered at the origin (0, 0) of a coordinate plane.
- Angles are measured counterclockwise from the positive x-axis.
- The unit circle is used to visualize and calculate trigonometric functions.
Exact Values
- sin(0°) = 0, cos(0°) = 1, and tan(0°) = 0.
- sin(30°) = 1/2, cos(30°) = √3/2, and tan(30°) = 1/√3.
- sin(45°) = 1/√2, cos(45°) = 1/√2, and tan(45°) = 1.
- sin(60°) = √3/2, cos(60°) = 1/2, and tan(60°) = √3.
Trigonometric Functions
- Sine (sin): Ratio of opposite side to hypotenuse in a right-angled triangle
- Formula: sin(A) = opposite side / hypotenuse
- Range: -1 ≤ sin(A) ≤ 1
- Period: 2π (sin(A + 2π) = sin(A))
-
Identities:
- sin(A) = cos(π/2 - A)
- sin(-A) = -sin(A)
Cosine (cos)
- Definition: Ratio of adjacent side to hypotenuse in a right-angled triangle
- Formula: cos(A) = adjacent side / hypotenuse
- Range: -1 ≤ cos(A) ≤ 1
- Period: 2π (cos(A + 2π) = cos(A))
-
Identities:
- cos(A) = sin(π/2 - A)
- cos(-A) = cos(A)
Tangent (tan)
- Definition: Ratio of sine to cosine of an angle
- Formula: tan(A) = sin(A) / cos(A)
- Range: All real numbers (-∞ to ∞)
- Period: π (tan(A + π) = tan(A))
-
Identities:
- tan(A) = -tan(-A)
- tan(A) = 1 / tan(π/2 - A)
Relationships between sine, cosine, and tangent
- Pythagorean Identity: sin^2(A) + cos^2(A) = 1
- Tan in terms of sin and cos: tan(A) = sin(A) / cos(A)
-
sin and cos in terms of tan:
- sin(A) = tan(A) / sqrt(1 + tan^2(A))
- cos(A) = 1 / sqrt(1 + tan^2(A))
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Description
Learn about trigonometric functions of special angles, including 30-60-90 and 45-45-90 triangles, and their reference angles. Understand the relationships between sides and angles in these triangles.
