Questions and Answers
Which of the following are methods of human reproduction?
What is the value of sin(30°)?
The cosine function is defined as the ratio of the opposite side to the hypotenuse.
False
What is the period of the sine wave?
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The cosecant function is the reciprocal of the ______ function.
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Match the following trigonometric functions with their definitions:
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The tangent function has a period of 2π.
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Which of the following angles has coordinates (0, 1) on the unit circle?
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What is the fourth key angle in degrees and radians where sine equals 1?
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Which identity states that 1 + tan² θ = sec² θ?
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The Pythagorean identity states that sin² θ + cos² θ = ______.
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Study Notes
Trigonometric Functions Overview
- Trigonometric functions connect angles with side ratios in right triangles.
- Key functions include sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).
Basic Trigonometric Functions
- Sine (sin θ): Ratio of the opposite side to the hypotenuse.
- Cosine (cos θ): Ratio of the adjacent side to the hypotenuse.
- Tangent (tan θ): Ratio of the opposite side to the adjacent side.
- Cosecant (csc θ): Reciprocal of sine, expressed as 1/sin θ.
- Secant (sec θ): Reciprocal of cosine, expressed as 1/cos θ.
- Cotangent (cot θ): Reciprocal of tangent, expressed as 1/tan θ.
The Unit Circle
- Defined as a circle with a radius of 1, centered at (0,0).
- Any point on the unit circle can be represented as (cos θ, sin θ).
- Angles are measured in radians; 360° equals 2π radians.
Key Angles and Coordinates
- 0° (0): (1, 0)
- 30° (π/6): (√3/2, 1/2)
- 45° (π/4): (√2/2, √2/2)
- 60° (π/3): (1/2, √3/2)
- 90° (π/2): (0, 1)
- 180° (π): (-1, 0)
- 270° (3π/2): (0, -1)
- 360° (2π): (1, 0)
Graphing Trigonometric Functions
- Sine Wave: Starts at (0,0), ranges from -1 to 1, period of 2π.
- Cosine Wave: Starts at (0,1), ranges from -1 to 1, period of 2π.
- Tangent Function: Period of π, has vertical asymptotes where cos θ = 0.
Trigonometric Identities
-
Pythagorean Identities:
- sin² θ + cos² θ = 1
- 1 + tan² θ = sec² θ
- 1 + cot² θ = csc² θ
-
Reciprocal Identities:
- csc θ = 1/sin θ
- sec θ = 1/cos θ
- cot θ = 1/tan θ
-
Angle Sum and Difference Identities:
- 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
Applications of Trigonometric Functions
- Used for solving triangles to find unknown sides and angles.
- Useful in modeling periodic phenomena such as sound waves.
- Essential in engineering and physics for solving problems involving angles and distances.
Inverse Trigonometric Functions
- Returns the angle for a given ratio.
- Notations include arcsin, arccos, arctan, and others.
- Each function has specific range and domain considerations.
Properties of Trigonometric Functions
- All trigonometric functions exhibit periodic behavior with defined periods.
- Functions show symmetry; cosine is even while sine is odd.
- Mastering these concepts is crucial for advanced studies in calculus and analytical geometry.
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Description
Test your knowledge on the various methods of human reproduction. This quiz covers key concepts and distinctions related to reproductive techniques. See how well you understand the different ways humans reproduce.
