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Questions and Answers
Given a right triangle with an acute angle of t, what is the formula for sine?
Given a right triangle with an acute angle of t, what is the formula for sine?
sint = opposite/hypotenuse
Given a right triangle with an acute angle of t, what is the formula for cosecant?
Given a right triangle with an acute angle of t, what is the formula for cosecant?
csc t = hypotenuse/opposite
What is the value of cos a in the triangle shown in Figure 3, where the adjacent side is 15 and the hypotenuse is 17?
What is the value of cos a in the triangle shown in Figure 3, where the adjacent side is 15 and the hypotenuse is 17?
cos a = 15/17
What is the value of sin t in the triangle shown in Figure 4, where the opposite side is 7 and the hypotenuse is 25?
What is the value of sin t in the triangle shown in Figure 4, where the opposite side is 7 and the hypotenuse is 25?
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In the triangle shown in Figure 6, what is the value of sin a given that the opposite side is 3 and the hypotenuse is 5?
In the triangle shown in Figure 6, what is the value of sin a given that the opposite side is 3 and the hypotenuse is 5?
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In the triangle shown in Figure 6, what is the value of cos a given that the adjacent side is 4 and the hypotenuse is 5?
In the triangle shown in Figure 6, what is the value of cos a given that the adjacent side is 4 and the hypotenuse is 5?
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In the triangle shown in Figure 6, what is the value of tan a given that the opposite side is 3 and the adjacent side is 4?
In the triangle shown in Figure 6, what is the value of tan a given that the opposite side is 3 and the adjacent side is 4?
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In the triangle shown in Figure 6, what is the value of csc a given that the hypotenuse is 5 and the opposite side is 3?
In the triangle shown in Figure 6, what is the value of csc a given that the hypotenuse is 5 and the opposite side is 3?
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In the triangle shown in Figure 6, what is the value of sec a given that the hypotenuse is 5 and the adjacent side is 4?
In the triangle shown in Figure 6, what is the value of sec a given that the hypotenuse is 5 and the adjacent side is 4?
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In the triangle shown in Figure 6, what is the value of cot a given that the adjacent side is 4 and the opposite side is 3?
In the triangle shown in Figure 6, what is the value of cot a given that the adjacent side is 4 and the opposite side is 3?
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What is the exact value of sin π/3 using side lengths?
What is the exact value of sin π/3 using side lengths?
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If sin t = 5/12, what is the value of cos (-t)?
If sin t = 5/12, what is the value of cos (-t)?
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If csc t = 2, what is the value of sec (-t)?
If csc t = 2, what is the value of sec (-t)?
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In the triangle shown in Figure 11, what is the length of the side opposite angle a, given that the hypotenuse is 14 and the angle a is 30 degrees (π/6 radians)?
In the triangle shown in Figure 11, what is the length of the side opposite angle a, given that the hypotenuse is 14 and the angle a is 30 degrees (π/6 radians)?
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In the triangle shown in Figure 11, what is the length of the side adjacent to angle a, given that the hypotenuse is 14 and the angle a is 30 degrees (π/6 radians)?
In the triangle shown in Figure 11, what is the length of the side adjacent to angle a, given that the hypotenuse is 14 and the angle a is 30 degrees (π/6 radians)?
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A right triangle has one angle of π/3 (60 degrees) and a hypotenuse of 20. What is the length of the side opposite the angle of π/3?
A right triangle has one angle of π/3 (60 degrees) and a hypotenuse of 20. What is the length of the side opposite the angle of π/3?
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A right triangle has one angle of π/3 (60 degrees) and a hypotenuse of 20. What is the length of the side adjacent to the angle of π/3?
A right triangle has one angle of π/3 (60 degrees) and a hypotenuse of 20. What is the length of the side adjacent to the angle of π/3?
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A person is standing 30 feet away from a tree. They measure the angle of elevation to the top of the tree as 57 degrees. What is the height of the tree?
A person is standing 30 feet away from a tree. They measure the angle of elevation to the top of the tree as 57 degrees. What is the height of the tree?
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A ladder leans against a building making an angle of 5π/12 radians with the ground. The ladder is 50 feet long and reaches a windowsill. What is the height of the windowsill?
A ladder leans against a building making an angle of 5π/12 radians with the ground. The ladder is 50 feet long and reaches a windowsill. What is the height of the windowsill?
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Study Notes
Trigonometric Functions
- Trigonometric functions relate angles in a right triangle to ratios of side lengths.
- Sine (sin): Opposite side / Hypotenuse
- Cosine (cos): Adjacent side / Hypotenuse
- Tangent (tan): Opposite side / Adjacent side
Unit Circle
- A triangle inscribed in a unit circle helps evaluate trigonometric functions.
- Coordinates (x, y) represent cosine (x) and sine (y) values.
- Angle (t), represented by a line segment connecting the origin to a point on the unit circle.
Evaluating Trigonometric Functions of Right Triangles
- Given a right triangle with an acute angle t, trigonometric functions are:
- sin t = opposite/hypotenuse
- cos t = adjacent/hypotenuse
- tan t = opposite/adjacent
- Example: In a triangle with hypotenuse 17 and adjacent side 15, cos α = 15/17
Cofunction Identities
- Sine and cosine are cofunctions, so sin θ = cos (90° - θ)
- Secant, cosecant, and cotangent have similar cofunction relationships.
Missing Side Lengths (Trigonometric Ratios)
- Use trigonometric ratios (sine, cosine, tangent) to find unknown sides in right triangles.
- Example: Given a hypotenuse of 20 and an angle of π/3, find other sides using sine/cosine/tangent relationships.
Measuring Distances Indirectly (Applications)
- Use trigonometric functions—often tangent—to find heights or distances that are hard to measure directly.
- Example: Find the height of a tree if the angle of elevation to the top of the tree from a point 30 ft away is 57°.
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Description
Test your knowledge of trigonometric functions and their relationships in right triangles. This quiz covers topics such as sine, cosine, tangent, and the unit circle, along with cofunction identities. Prepare to apply these concepts to various problems and examples.
