Podcast
Questions and Answers
In a right triangle, the cosine of an angle is equal to the ratio of the side opposite the angle to the hypotenuse.
In a right triangle, the cosine of an angle is equal to the ratio of the side opposite the angle to the hypotenuse.
False
The Pythagorean Theorem states that for any right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse.
The Pythagorean Theorem states that for any right triangle, the sum of the squares of the two legs is equal to the square of the hypotenuse.
True
The tangent of an angle is defined as the ratio of the sides adjacent to the angle.
The tangent of an angle is defined as the ratio of the sides adjacent to the angle.
False
Cosecant, secant, and cotangent are the reciprocal trigonometric functions of sine, cosine, and tangent, respectively.
Cosecant, secant, and cotangent are the reciprocal trigonometric functions of sine, cosine, and tangent, respectively.
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Sin²θ + Cos²θ = 1 is an example of a trigonometric identity that holds true for all angles.
Sin²θ + Cos²θ = 1 is an example of a trigonometric identity that holds true for all angles.
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Angles larger than 90° do not have defined trigonometric values.
Angles larger than 90° do not have defined trigonometric values.
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Trigonometry has no applications in fields such as physics or computer graphics.
Trigonometry has no applications in fields such as physics or computer graphics.
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The unit circle has a radius of zero and is centered at the origin.
The unit circle has a radius of zero and is centered at the origin.
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Study Notes
Introduction to Trigonometry
- Trigonometry is a branch of mathematics that studies relationships between angles and sides of triangles.
- Primarily concerned with calculating angles and lengths of right-angled triangles.
- Fundamental concepts include sine, cosine, and tangent.
Trigonometric Ratios
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Sine (sin): Represents the ratio of the side opposite to the angle to the hypotenuse.
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sin θ = opposite / hypotenuse
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Cosine (cos): Represents the ratio of the side adjacent to the angle to the hypotenuse.
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cos θ = adjacent / hypotenuse
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Tangent (tan): Represents the ratio of the side opposite to the angle to the side adjacent to the angle.
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tan θ = opposite / adjacent
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Pythagorean Theorem
- States the relationship between the sides of a right-angled triangle:
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a² + b² = c², whereaandbare the legs andcis the hypotenuse.
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Reciprocal Trigonometric Functions
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Cosecant (csc):
csc θ = 1/sin θ -
Secant (sec):
sec θ = 1/cos θ -
Cotangent (cot):
cot θ = 1/tan θ
Trigonometric Identities
- Equations that hold true for all angles for which both sides of the equation are defined.
- Example:
Sin²θ + Cos²θ = 1
Angles and their Trigonometric values
- Specific angles (e.g., 0°, 30°, 45°, 60°, 90°) have known and exact trigonometric values.
- These values are commonly used in calculations and are often memorized.
Applications of Trigonometry
- Navigation: Determining distances and directions.
- Engineering: Calculating heights and distances in construction and surveying.
- Physics: Analyzing forces and motion in objects.
- Astronomy: Calculating distances and positions of celestial bodies.
- Computer Graphics: Creating realistic images and 3D models (e.g., rendering systems).
Trigonometric Functions of General Angles
- Trigonometric functions can be extended to angles larger than 90° or negative angles by considering the unit circle.
- Quadrants are used to determine the signs (+ or -) of the trigonometric functions for angles in different positions.
Solving Right Triangles
- Using the trigonometric ratios and Pythagorean Theorem, unknown sides and angles can be determined in a right-angled triangle with given information.
- Given one angle and one side: Calculate the other sides.
- Given two sides: Calculate an angle and other side.
Unit Circle
- A circle with a radius of one unit, centred at the origin.
- Used to define trigonometric functions for any angle.
- Points on the circle relate to the trigonometric values through their coordinates.
Trigonometric Graphs
- Sine, cosine, and tangent functions have specific graphs.
- The graphs show periodic behaviour with certain amplitudes and periods. Understanding these graphs is crucial for visualizing trigonometric functions and identifying patterns.
Inverse Trigonometric Functions
- Functions that "undo" the trigonometric functions.
- Used to find the angle when the trigonometric ratio is known.
- Examples include arcsin, arccos, arctan.
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Description
Explore the fundamental concepts of trigonometry, including trigonometric ratios like sine, cosine, and tangent. Delve into the Pythagorean theorem and learn about reciprocal trigonometric functions. This quiz is essential for understanding the relationships between angles and sides in triangles.
