Podcast
Questions and Answers
What is the relationship between the tangent and sine and cosine functions?
What is the relationship between the tangent and sine and cosine functions?
Which trigonometric ratio represents the reciprocal of cosine?
Which trigonometric ratio represents the reciprocal of cosine?
Which trigonometric ratio is defined as the ratio of the length of the side adjacent to an angle to the length of the hypotenuse?
Which trigonometric ratio is defined as the ratio of the length of the side adjacent to an angle to the length of the hypotenuse?
What is the Pythagorean identity relating sine and cosine?
What is the Pythagorean identity relating sine and cosine?
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In a right-angled triangle, if the length of the side opposite an angle is 5 units and the length of the hypotenuse is 13 units, what is the value of the sine of that angle?
In a right-angled triangle, if the length of the side opposite an angle is 5 units and the length of the hypotenuse is 13 units, what is the value of the sine of that angle?
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Given a right-angled triangle with an acute angle of 30 degrees, what is the relationship between the lengths of the opposite and adjacent sides?
Given a right-angled triangle with an acute angle of 30 degrees, what is the relationship between the lengths of the opposite and adjacent sides?
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Which scenario would NOT require the use of trigonometric ratios for solving?
Which scenario would NOT require the use of trigonometric ratios for solving?
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If the cotangent (cot θ) of an angle is 3/4, what is the value of the tangent (tan θ)?
If the cotangent (cot θ) of an angle is 3/4, what is the value of the tangent (tan θ)?
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Flashcards
Sine (sin)
Sine (sin)
Ratio of opposite side to hypotenuse in a triangle. sin θ = opposite / hypotenuse
Cosine (cos)
Cosine (cos)
Ratio of adjacent side to hypotenuse in a triangle. cos θ = adjacent / hypotenuse
Tangent (tan)
Tangent (tan)
Ratio of opposite side to adjacent side in a triangle. tan θ = opposite / adjacent
Cosecant (csc)
Cosecant (csc)
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Secant (sec)
Secant (sec)
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Cotangent (cot)
Cotangent (cot)
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Pythagorean Identity
Pythagorean Identity
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Trigonometric Applications
Trigonometric Applications
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Study Notes
Basic Trigonometric Ratios
- Trigonometric ratios relate the angles of a right-angled triangle to the ratios of its side lengths. They are fundamental in trigonometry.
- The most common trigonometric ratios are sine (sin), cosine (cos), and tangent (tan).
- These ratios are defined for an acute angle θ in a right-angled triangle.
- Sine (sin θ): The ratio of the length of the side opposite to the angle θ to the length of the hypotenuse. sin θ = opposite / hypotenuse
- Cosine (cos θ): The ratio of the length of the side adjacent to the angle θ to the length of the hypotenuse. cos θ = adjacent / hypotenuse
- Tangent (tan θ): The ratio of the length of the side opposite to the angle θ to the length of the side adjacent to the angle θ. tan θ = opposite / adjacent
Reciprocal Trigonometric Ratios
- There are three reciprocal trigonometric ratios: cosecant (csc), secant (sec), and cotangent (cot).
- Cosecant (csc θ): The reciprocal of sine. csc θ = 1 / sin θ
- Secant (sec θ): The reciprocal of cosine. sec θ = 1 / cos θ
- Cotangent (cot θ): The reciprocal of tangent. cot θ = 1 / tan θ
Important Relationships
- The Pythagorean identity relates the sine and cosine functions: sin²θ + cos²θ = 1
- This identity holds true for all angles θ.
- Using the definitions of sine, cosine and tangent, and the Pythagorean Theorem, we can derive the following relationships:
- tan θ = sin θ / cos θ
- cot θ = cos θ / sin θ
Usage in Solving Right-Angled Triangles
- Trigonometric ratios are used to find unknown side lengths and angles in right-angled triangles.
- Given one side length and an acute angle, you can find other side lengths and angles using the appropriate ratios.
- The choice of which ratio to use depends on the known and unknown values.
- Example problems may involve finding the length of a missing side or the measure of a missing angle using a known side and a known angle, or two known sides.
Beyond Right-Angled Triangles
- Trigonometric ratios can be extended to obtuse angles and angles larger than 90 degrees.
- This involves incorporating the trigonometric definitions into specific quadrants in the coordinate plane, enabling the use of values for angles outside the 0° to 90° range.
Applications
- Trigonometric ratios are used in many fields, including:
- Engineering: Designing structures, calculating distances, navigating.
- Navigation: Determining bearings, plotting courses.
- Surveying: Measuring land areas, determining heights.
- Physics: Calculating forces, velocities, and other physical quantities.
- A practical application is determining the height of a building using the angle of elevation and the distance from the building to the observer.
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Description
Explore the foundational concepts of trigonometric ratios in this quiz. You will learn about sine, cosine, tangent, and their reciprocal functions. Understand how these ratios relate to the angles and sides of right-angled triangles.
