Podcast
Questions and Answers
What is the definition of cosine in a right-angled triangle?
What is the definition of cosine in a right-angled triangle?
Which of the following statements about the Pythagorean identity is true?
Which of the following statements about the Pythagorean identity is true?
Which trigonometric ratio is the reciprocal of the sine function?
Which trigonometric ratio is the reciprocal of the sine function?
What does the unit circle help to represent in trigonometry?
What does the unit circle help to represent in trigonometry?
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Which of the following identities is true for the tangent function?
Which of the following identities is true for the tangent function?
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In which of the following applications is trigonometry crucial?
In which of the following applications is trigonometry crucial?
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How can trigonometric equations be solved effectively?
How can trigonometric equations be solved effectively?
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What is the relationship between sine and cosine for any angle θ as defined by the unit circle?
What is the relationship between sine and cosine for any angle θ as defined by the unit circle?
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Study Notes
Introduction to Trigonometry
- Trigonometry studies relationships between angles and sides of triangles.
- Applications include navigation, surveying, engineering, and physics.
Basic Trigonometric Ratios
- Fundamental ratios are sine (sin), cosine (cos), and tangent (tan).
- Defined for acute angles in right-angled triangles.
- sin θ = opposite side / hypotenuse
- cos θ = adjacent side / hypotenuse
- tan θ = opposite side / adjacent side
Other Trigonometric Ratios
- Cosecant (csc θ) is the reciprocal of sine (csc θ = 1/sin θ).
- Secant (sec θ) is the reciprocal of cosine (sec θ = 1/cos θ).
- Cotangent (cot θ) is the reciprocal of tangent (cot θ = 1/tan θ).
Relationship Between Trigonometric Ratios
- Pythagorean identity: sin²θ + cos²θ = 1.
Trigonometric Functions for Any Angle
- Extended to any angle using the unit circle.
- Unit circle is a circle with radius 1 centered at the origin.
- X-coordinate represents cosine of angle, y-coordinate represents sine.
- Tangent of an angle is the ratio of sine to cosine (tan θ = sin θ / cos θ).
Trigonometric Identities
- Equations true for all valid values.
- Key identities include:
- sin(-θ) = -sinθ
- cos(-θ) = cosθ
- tan(-θ) = -tanθ
- Used to simplify expressions and solve equations.
Trigonometric Equations
- Involve trigonometric functions, solved using manipulation, identities, and unit circle.
- May have multiple solutions due to periodicity.
Applications of Trigonometry
- Crucial for solving triangle-related problems.
- Applications include:
- Calculating heights and distances (surveying).
- Determining angles (navigation).
- Analyzing object motion (physics).
- Designing structures (engineering).
Graphs of Trigonometric Functions
- Periodic functions repeating at intervals.
- Sine, cosine, and tangent graphs show characteristic waves.
- Sine and cosine period is 360° (or 2π radians).
- Tangent period is 180° (or π radians).
Inverse Trigonometric Functions
- Used to find angles given trigonometric ratios.
- Notations include arcsin (sin⁻¹), arccos (cos⁻¹), arctan (tan⁻¹), etc.
- Produce angles within specific ranges.
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Description
This quiz covers the basics of trigonometry, including fundamental trigonometric ratios such as sine, cosine, and tangent. It also explores other ratios and their relationships, emphasizing the importance of these concepts in various applications like engineering and physics.
