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Questions and Answers
In a right-angled triangle, what is the ratio that defines the sine (sin) of an angle $\theta$?
In a right-angled triangle, what is the ratio that defines the sine (sin) of an angle $\theta$?
What is the value of $\csc \theta$ if $\sin \theta = 0.5$?
What is the value of $\csc \theta$ if $\sin \theta = 0.5$?
Which trigonometric identity is represented by $\sin^2 \theta + \cos^2 \theta = 1$?
Which trigonometric identity is represented by $\sin^2 \theta + \cos^2 \theta = 1$?
What are the coordinates of the point where the terminal side of an angle $\theta$ intersects the unit circle?
What are the coordinates of the point where the terminal side of an angle $\theta$ intersects the unit circle?
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What is the period of the sine function, $\sin(x)$?
What is the period of the sine function, $\sin(x)$?
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What is the range of the cosine function?
What is the range of the cosine function?
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Which function is derived by dividing the $\sin \theta$ by $\cos \theta$?
Which function is derived by dividing the $\sin \theta$ by $\cos \theta$?
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The tangent function has vertical asymptotes at which points?
The tangent function has vertical asymptotes at which points?
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Study Notes
Introduction to Trigonometric Functions
- Trigonometric functions relate angles in a right-angled triangle to ratios of side lengths.
- These functions are fundamental in mathematics, physics, and engineering.
- Common trigonometric functions include sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).
Definitions in a Right-Angled Triangle
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Consider a right-angled triangle with:
- Angle θ
- Opposite side (opposite to θ)
- Adjacent side (adjacent to θ and excluding the hypotenuse)
- Hypotenuse (the longest side, opposite the right angle)
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Sine (sin θ) = Opposite / Hypotenuse
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Cosine (cos θ) = Adjacent / Hypotenuse
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Tangent (tan θ) = Opposite / Adjacent
Definitions Using the Unit Circle
- Define the trigonometric functions using the unit circle approach.
- The unit circle is a circle with a radius of 1 centered at the origin (0,0) of a Cartesian plane.
- An angle θ in standard position has its initial side on the positive x-axis.
- The point where the terminal side intersects the unit circle has coordinates (cos θ, sin θ).
Relationships Between Trigonometric Functions
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Reciprocal Identities:
- csc θ = 1 / sin θ
- sec θ = 1 / cos θ
- cot θ = 1 / tan θ = cos θ / sin θ
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Pythagorean Identities:
- sin² θ + cos² θ = 1
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Tangent Identity:
- tan θ = sin θ / cos θ
Trigonometric Functions of Special Angles
- Knowledge of trigonometric functions for special angles (0°, 30°, 45°, 60°, 90°) is crucial.
- These values are often memorized or readily derived from geometric relationships in special triangles.
Graphs of Trigonometric Functions
- Sine function (sin x): periodic function that oscillates between -1 and 1, with a period of 2π.
- Cosine function (cos x): similar to the sine function, but shifted horizontally by π/2.
- Tangent function (tan x): periodic function with vertical asymptotes at odd multiples of π/2.
Domain and Range of Trigonometric Functions
- Domain: the set of all possible input values (angles) for a function.
- Range: the set of all possible output values for a function.
- The domains and ranges are defined differently for each trigonometric function and can include restrictions.
Applications of Trigonometric Functions
- Trigonometry is used in navigation, surveying, engineering, and many other fields.
- Essential for modeling periodic phenomena, particularly in areas like sound and light waves, and harmonic motion.
Unit Circle and Angles
- The unit circle is a valuable tool for visualizing trigonometric values for any angle.
- Angles can be measured in degrees or radians.
- Conversion between degree and radian measure is important.
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Description
This quiz covers the foundational concepts of trigonometric functions, including definitions related to right-angled triangles and the unit circle. You'll explore key functions such as sine, cosine, and tangent, and how they relate to the geometry of triangles. Test your understanding of these essential mathematical functions.
