Questions and Answers
What type of triangle has one angle greater than 90 degrees?
All angles in an acute triangle are less than 90 degrees.
True
What is the sine of an angle in a right triangle?
The ratio of the opposite side to the hypotenuse.
The Pythagorean identity states that sin²(θ) + cos²(θ) = _____
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Match the trigonometric functions with their definitions:
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What is the cosine of an angle in a right triangle?
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The range of the tangent function is limited to values between -1 and 1.
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Convert 180 degrees into radians.
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Study Notes
Basic Concepts
- Definition: Study of relationships between angles and sides of triangles.
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Types of Triangles:
- Right Triangle: One angle is 90 degrees.
- Acute Triangle: All angles are less than 90 degrees.
- Obtuse Triangle: One angle is greater than 90 degrees.
Key Functions
- Sine (sin): Ratio of the opposite side to the hypotenuse.
- Cosine (cos): Ratio of the adjacent side to the hypotenuse.
- Tangent (tan): Ratio of the opposite side to the adjacent side.
Basic Identities
-
Pythagorean Identity:
- ( \sin^2(\theta) + \cos^2(\theta) = 1 )
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Reciprocal Identities:
- ( \csc(\theta) = \frac{1}{\sin(\theta)} )
- ( \sec(\theta) = \frac{1}{\cos(\theta)} )
- ( \cot(\theta) = \frac{1}{\tan(\theta)} )
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Quotient Identities:
- ( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} )
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Even-Odd Identities:
- ( \sin(-\theta) = -\sin(\theta) )
- ( \cos(-\theta) = \cos(\theta) )
Angle Measures
-
Degrees and Radians:
- ( 180^\circ = \pi ) radians
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Common Angles:
- ( 0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ )
Unit Circle
- Definition: Circle with a radius of 1 centered at the origin.
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Key Points:
- ( (1, 0) ) at ( 0^\circ )
- ( (0, 1) ) at ( 90^\circ )
- ( (-1, 0) ) at ( 180^\circ )
- ( (0, -1) ) at ( 270^\circ )
Graphs of Trigonometric Functions
-
Sine Function:
- Period: ( 2\pi )
- Range: ([-1, 1])
-
Cosine Function:
- Period: ( 2\pi )
- Range: ([-1, 1])
-
Tangent Function:
- Period: ( \pi )
- Range: All real numbers
Applications
-
Solving Triangles:
- Methods to find unknown sides and angles using trigonometric ratios.
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Real-World Applications:
- Physics (waves, oscillations), Engineering, Architecture, Navigation.
Additional Concepts
-
Law of Sines:
- ( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} )
-
Law of Cosines:
- ( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) )
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Inverse Trigonometric Functions:
- ( \sin^{-1}(x), \cos^{-1}(x), \tan^{-1}(x) ) for finding angles.
Basic Concepts
- Trigonometry studies the relationships between angles and sides of triangles.
- Types of triangles include:
- Right Triangle: Contains one angle of 90 degrees.
- Acute Triangle: All angles are less than 90 degrees.
- Obtuse Triangle: One angle is greater than 90 degrees.
Key Functions
- Sine (sin): The ratio of the length of the opposite side to the hypotenuse.
- Cosine (cos): The ratio of the length of the adjacent side to the hypotenuse.
- Tangent (tan): The ratio of the length of the opposite side to the adjacent side.
Basic Identities
- Pythagorean Identity: ( \sin^2(\theta) + \cos^2(\theta) = 1 ).
- Reciprocal Identities:
- Cosecant: ( \csc(\theta) = \frac{1}{\sin(\theta)} ).
- Secant: ( \sec(\theta) = \frac{1}{\cos(\theta)} ).
- Cotangent: ( \cot(\theta) = \frac{1}{\tan(\theta)} ).
- Quotient Identity: ( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} ).
- Even-Odd Identities:
- ( \sin(-\theta) = -\sin(\theta) ).
- ( \cos(-\theta) = \cos(\theta) ).
Angle Measures
- Angle conversion between degrees and radians: ( 180^\circ = \pi ) radians.
- Common angle values: ( 0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ ).
Unit Circle
- The unit circle has a radius of 1 and is centered at the origin.
- Key points on the unit circle include:
- ( (1, 0) ) corresponds to ( 0^\circ ).
- ( (0, 1) ) corresponds to ( 90^\circ ).
- ( (-1, 0) ) corresponds to ( 180^\circ ).
- ( (0, -1) ) corresponds to ( 270^\circ ).
Graphs of Trigonometric Functions
- Sine Function:
- Period is ( 2\pi ).
- Range is from -1 to 1.
- Cosine Function:
- Period is ( 2\pi ).
- Range is from -1 to 1.
- Tangent Function:
- Period is ( \pi ).
- Range includes all real numbers.
Applications
- Solving triangles involves finding unknown sides and angles using trigonometric ratios.
- Real-world applications span across fields like physics (waves, oscillations), engineering, architecture, and navigation.
Additional Concepts
- Law of Sines: Relates the sides and angles of a triangle through the formula ( \frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)} ).
- Law of Cosines: Used to calculate a side of a triangle with ( c^2 = a^2 + b^2 - 2ab \cdot \cos(C) ).
- Inverse Trigonometric Functions: Functions like ( \sin^{-1}(x), \cos^{-1}(x), \tan^{-1}(x) ) are utilized for calculating angles from known ratios.
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Description
This quiz explores the basic concepts of trigonometry, focusing on the relationships between angles and sides of triangles. It includes definitions, types of triangles, key functions like sine, cosine, and tangent, as well as essential trigonometric identities. Test your understanding of these fundamental principles!
