Questions and Answers
What is the definition of trigonometry and its main focus?
Trigonometry is the study of relationships between the angles and sides of triangles, particularly right triangles.
Explain the relationship described by the Pythagorean Theorem.
The Pythagorean Theorem states that in a right triangle, the square of the length of the hypotenuse ($c$) is equal to the sum of the squares of the lengths of the other two sides ($a^2 + b^2 = c^2$).
What is the significance of the unit circle in trigonometry?
The unit circle is a circle with a radius of 1 centered at the origin, which helps to define the sine and cosine functions as coordinates on the circle for key angles.
Provide the Pythagorean identity involving sine and cosine.
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Describe the periodic nature of sine and cosine functions.
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How does the Law of Sines relate sides and angles in a triangle?
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What functions are included in the category of inverse trigonometric functions?
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State the law of cosines and its relevance.
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Study Notes
Trigonometry
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Definition: Trigonometry is the study of relationships between the angles and sides of triangles, especially right triangles.
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Basic Functions:
- Sine (sin): Opposite side / Hypotenuse
- Cosine (cos): Adjacent side / Hypotenuse
- Tangent (tan): Opposite side / Adjacent side
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Reciprocal Functions:
- Cosecant (csc) = 1/sin
- Secant (sec) = 1/cos
- Cotangent (cot) = 1/tan
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Right Triangle Relationships:
- Pythagorean Theorem: a² + b² = c² (where c is the hypotenuse)
- Angles: The sum of angles in a triangle is 180°.
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Unit Circle:
- A circle with a radius of 1 centered at the origin of a coordinate plane.
- Key angles in radians:
- 0 (0°), π/6 (30°), π/4 (45°), π/3 (60°), π/2 (90°), π (180°)
- Coordinates on the unit circle represent (cos θ, sin θ).
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Trigonometric Identities:
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Pythagorean Identities:
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = csc²θ
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Angle 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
- tan(a ± b) = (tan a ± tan b) / (1 ∓ tan a tan b)
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Pythagorean Identities:
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Graphs of Trigonometric Functions:
- Sine and Cosine: Periodic with period 2π, amplitude 1.
- Tangent: Period π, undefined at (π/2 + nπ).
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Applications:
- Used in physics for wave motion, oscillations, and in engineering for structural analysis.
- Important in navigation, astronomy, and computer graphics.
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Inverse Trigonometric Functions:
- Used to find angles when the values of trigonometric functions are known.
- Functions: arcsin, arccos, arctan, etc.
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Law of Sines:
- a/sin A = b/sin B = c/sin C (relates sides and angles in any triangle).
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Law of Cosines:
- c² = a² + b² - 2ab * cos C (generalizes the Pythagorean theorem).
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Trigonometric Ratios for Common Angles:
- 0°: sin 0 = 0, cos 0 = 1, tan 0 = 0
- 30°: sin 30 = 1/2, cos 30 = √3/2, tan 30 = 1/√3
- 45°: sin 45 = √2/2, cos 45 = √2/2, tan 45 = 1
- 60°: sin 60 = √3/2, cos 60 = 1/2, tan 60 = √3
- 90°: sin 90 = 1, cos 90 = 0, tan 90 = undefined
Trigonometry Overview
- Study of relationships between angles and sides in triangles, particularly right triangles.
Basic Trigonometric Functions
- Sine (sin): Ratio of the length of the opposite side to the hypotenuse.
- Cosine (cos): Ratio of the length of the adjacent side to the hypotenuse.
- Tangent (tan): Ratio of the length of the opposite side to the adjacent side.
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Reciprocal Functions:
- Cosecant (csc): 1/sin
- Secant (sec): 1/cos
- Cotangent (cot): 1/tan
Right Triangle Relationships
- Pythagorean Theorem: a² + b² = c², where c is the hypotenuse.
- The sum of angles in any triangle is always 180°.
Unit Circle
- A circle with a radius of 1, centered at the origin of a coordinate plane.
- Key angles in radians include:
- 0 (0°)
- π/6 (30°)
- π/4 (45°)
- π/3 (60°)
- π/2 (90°)
- π (180°)
- Coordinates on the unit circle are represented as (cos θ, sin θ).
Trigonometric Identities
-
Pythagorean Identities:
- sin²θ + cos²θ = 1
- 1 + tan²θ = sec²θ
- 1 + cot²θ = csc²θ
-
Angle 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
- tan(a ± b) = (tan a ± tan b) / (1 ∓ tan a tan b)
Graphs of Trigonometric Functions
- Sine and Cosine Functions: Both functions are periodic with a period of 2π and an amplitude of 1.
- Tangent Function: Period of π with undefined values at (π/2 + nπ).
Applications
- Trigonometry is fundamental in physics, especially in wave motion and oscillations.
- Engineering uses trigonometry for structural analysis.
- Important tools in navigation, astronomy, and computer graphics.
Inverse Trigonometric Functions
- Used to calculate angles when the values of trigonometric functions are known.
- Common functions include arcsin, arccos, and arctan.
Law of Sines
- a/sin A = b/sin B = c/sin C, relating the sides and angles of any triangle.
Law of Cosines
- Generalizes the Pythagorean theorem: c² = a² + b² - 2ab * cos C.
Trigonometric Ratios for Common Angles
- 0°: sin 0 = 0, cos 0 = 1, tan 0 = 0
- 30°: sin 30 = 1/2, cos 30 = √3/2, tan 30 = 1/√3
- 45°: sin 45 = √2/2, cos 45 = √2/2, tan 45 = 1
- 60°: sin 60 = √3/2, cos 60 = 1/2, tan 60 = √3
- 90°: sin 90 = 1, cos 90 = 0, tan 90 is undefined.
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Description
This quiz covers fundamental concepts of trigonometry, including basic functions, relationships in right triangles, and the unit circle. Test your knowledge on trigonometric identities and the Pythagorean theorem to solidify your understanding of this essential mathematical field.
