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Questions and Answers
Which function represents the ratio of the opposite side to the hypotenuse in a right triangle?
The cosine function is the reciprocal of the tangent function.
False
What is the Pythagorean identity in trigonometry?
sin²(θ) + cos²(θ) = 1
The tangent function can be expressed as the ratio of the ______ side to the ______ side.
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Match the function to its relationship:
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What is the sine of 30 degrees?
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The graph of the sine function oscillates between -1 and 1.
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State the law of cosines formula.
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Explain how to convert an angle from degrees to radians.
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What are the coordinates of a point on the unit circle at an angle of $\frac{\pi}{3}$ radians?
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Describe the relationship between sine and cosine using the Pythagorean identity.
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How does the tangent function behave as its input approaches $\frac{\pi}{2}$?
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What is the result of $\sin(2\theta)$ using the double angle formula?
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List the three reciprocal trigonometric functions and their definitions.
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Which trigonometric functions are used to calculate the height of an object based on the angle of elevation?
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Using the angle sum identity, express $\sin(45^\circ + 30^\circ)$ in terms of sine and cosine.
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Study Notes
Trigonometry
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Definition: Branch of mathematics dealing with the relationships between the angles and sides of triangles, particularly right triangles.
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Basic Functions:
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Sine (sin): Ratio of the opposite side to the hypotenuse.
- sin(θ) = Opposite/Hypotenuse
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Cosine (cos): Ratio of the adjacent side to the hypotenuse.
- cos(θ) = Adjacent/Hypotenuse
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Tangent (tan): Ratio of the opposite side to the adjacent side.
- tan(θ) = Opposite/Adjacent
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Sine (sin): Ratio of the opposite side to the hypotenuse.
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Reciprocal Functions:
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Cosecant (csc): Reciprocal of sine.
- csc(θ) = 1/sin(θ)
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Secant (sec): Reciprocal of cosine.
- sec(θ) = 1/cos(θ)
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Cotangent (cot): Reciprocal of tangent.
- cot(θ) = 1/tan(θ)
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Cosecant (csc): Reciprocal of sine.
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Pythagorean Identity:
- sin²(θ) + cos²(θ) = 1
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Angle Measures:
- Degrees: Full circle = 360 degrees.
- Radians: Full circle = 2π radians; 180 degrees = π radians.
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Special Angles:
- 0°, 30°, 45°, 60°, 90°
- Corresponding sine and cosine values:
- 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
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Trigonometric Identities:
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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)
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Double Angle Formulas:
- sin(2θ) = 2sin(θ)cos(θ)
- cos(2θ) = cos²(θ) - sin²(θ)
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Sum and Difference Formulas:
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Applications:
- Used in physics, engineering, architecture, and computer graphics.
- Essential for solving problems involving angles and distances.
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Graphs:
- Sin and cos functions oscillate between -1 and 1.
- Tan function has vertical asymptotes and repeats every π radians.
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Law of Sines:
- a/sin(A) = b/sin(B) = c/sin(C), where a, b, c are sides opposite to angles A, B, C respectively.
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Law of Cosines:
- c² = a² + b² - 2ab*cos(C), relating the lengths of the sides of a triangle to the cosine of one of its angles.
Trigonometry
- Definition: A mathematical branch focused on relationships between angles and sides in triangles, especially right triangles.
Basic Functions
- Sine (sin): Defines the ratio of the opposite side to the hypotenuse. Formula: sin(θ) = Opposite/Hypotenuse.
- Cosine (cos): Determines the ratio of the adjacent side to the hypotenuse. Formula: cos(θ) = Adjacent/Hypotenuse.
- Tangent (tan): Represents the ratio of the opposite side to the adjacent side. Formula: tan(θ) = Opposite/Adjacent.
Reciprocal Functions
- Cosecant (csc): The reciprocal of sine. Formula: csc(θ) = 1/sin(θ).
- Secant (sec): The reciprocal of cosine. Formula: sec(θ) = 1/cos(θ).
- Cotangent (cot): The reciprocal of tangent. Formula: cot(θ) = 1/tan(θ).
Pythagorean Identity
- Key relationship: sin²(θ) + cos²(θ) = 1.
Angle Measures
- Degrees: A full circle consists of 360 degrees.
- Radians: A full circle equals 2π radians; 180 degrees corresponds to π radians.
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Special Angles: Notable angles include 0°, 30°, 45°, 60°, and 90° with corresponding sine and cosine values:
- 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.
Trigonometric Identities
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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).
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Double Angle Formulas:
- sin(2θ) = 2sin(θ)cos(θ).
- cos(2θ) = cos²(θ) - sin²(θ).
Applications
- Widely used in physics, engineering, architecture, and computer graphics, particularly in angle and distance problem-solving.
Graphs
- Sine and cosine functions oscillate between -1 and 1.
- Tangent function features vertical asymptotes and repeats every π radians.
Laws of Triangles
- Law of Sines: a/sin(A) = b/sin(B) = c/sin(C), where a, b, c are the lengths of sides opposite angles A, B, C, respectively.
- Law of Cosines: c² = a² + b² - 2ab*cos(C), linking side lengths with the cosine of an angle.
Trigonometry Overview
- Study of relationships between the angles and sides of triangles, particularly in right triangles.
Key Functions
- Sine (sin): Measures the ratio of the length of the opposite side to the hypotenuse.
- Cosine (cos): Measures the ratio of the length of the adjacent side to the hypotenuse.
- Tangent (tan): Measures the ratio of the length of the opposite side to the adjacent side.
Reciprocal Functions
- Cosecant (csc): Inverse of sine, calculated as ( \csc(\theta) = \frac{1}{\sin(\theta)} ).
- Secant (sec): Inverse of cosine, calculated as ( \sec(\theta) = \frac{1}{\cos(\theta)} ).
- Cotangent (cot): Inverse of tangent, calculated as ( \cot(\theta) = \frac{1}{\tan(\theta)} ).
Pythagorean Identity
- Fundamental identity expressing the relationship between sine and cosine: ( \sin^2(\theta) + \cos^2(\theta) = 1 ).
Angles
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Degrees and Radians:
- Full circle: 360 degrees equals ( 2\pi ) radians.
- Conversion formula: ( \text{Degrees} = \text{Radians} \times \frac{180}{\pi} ).
Unit Circle
- A circle with a radius of 1, centered at the origin, essential for understanding trigonometric functions.
- Coordinates on the circle provide sine and cosine values at any angle ( \theta ): ( (x, y) = (\cos(\theta), \sin(\theta)) ).
Trigonometric Identities
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Angle Sum and Difference:
- Sine and cosine identities allow calculation of sine and cosine for combined angles.
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Double Angle Formulas:
- Express sine and cosine for twice an angle in terms of the original angle.
Applications
- Trigonometry plays a critical role in various fields including physics, engineering, and computer graphics.
- Key for solving problems involving oscillations, waves, and rotations.
Graphing
- Sine and cosine functions oscillate between -1 and 1, demonstrating periodic behavior.
- Tangent function features vertical asymptotes at points where ( \cos(\theta) = 0 ).
Solving Trigonometric Equations
- Techniques include manipulating equations algebraically, utilizing identities, and graphing functions for solutions.
Inverse Trigonometric Functions
- Used to determine angle measures from known ratios: ( \sin^{-1}(x) ), ( \cos^{-1}(x) ), and ( \tan^{-1}(x) ).
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Description
Test your understanding of trigonometry, focusing on basic functions, reciprocal functions, and angle measures. This quiz covers critical concepts such as sine, cosine, and the Pythagorean identity. Perfect for students looking to reinforce their knowledge in this essential branch of mathematics.
