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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?
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?
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?
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?
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?
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?
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?
What is the exact value of sin π/3 using side lengths?
What is the exact value of sin π/3 using side lengths?
If sin t = 5/12, what is the value of cos (-t)?
If sin t = 5/12, what is the value of cos (-t)?
If csc t = 2, what is the value of sec (-t)?
If csc t = 2, what is the value of sec (-t)?
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 (Ï€/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 (Ï€/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 (Ï€/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 (Ï€/6 radians)?
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?
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?
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?
A ladder leans against a building making an angle of 5Ï€/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 5Ï€/12 radians with the ground. The ladder is 50 feet long and reaches a windowsill. What is the height of the windowsill?
Flashcards
Sine (sin t)
Sine (sin t)
The ratio of the side opposite the angle to the hypotenuse in a right triangle.
Cosine (cos t)
Cosine (cos t)
The ratio of the side adjacent to the angle to the hypotenuse in a right triangle.
Tangent (tan t)
Tangent (tan t)
The ratio of the side opposite the angle to the side adjacent to the angle in a right triangle.
Secant (sec t)
Secant (sec t)
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Cosecant (csc t)
Cosecant (csc t)
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Cotangent (cot t)
Cotangent (cot t)
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Trigonometric Functions on Unit Circle
Trigonometric Functions on Unit Circle
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Complementary Angles
Complementary Angles
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Cofunction Identities
Cofunction Identities
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Standard Position Angle
Standard Position Angle
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Right Triangle in Standard Position
Right Triangle in Standard Position
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Special Angles
Special Angles
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Solving Right Triangles
Solving Right Triangles
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Applied Problems with Right Triangles
Applied Problems with Right Triangles
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Tangent of an angle
Tangent of an angle
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Cotangent of an angle
Cotangent of an angle
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Secant of an angle
Secant of an angle
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Cosecant of an angle
Cosecant of an angle
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Triangle Inscribed in Unit Circle
Triangle Inscribed in Unit Circle
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Trigonometric Values of Special Angles
Trigonometric Values of Special Angles
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30-60-90 Triangle Ratios
30-60-90 Triangle Ratios
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45-45-90 Triangle Ratios
45-45-90 Triangle Ratios
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Unit Circle
Unit Circle
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Exact Values of Trigonometric Functions
Exact Values of Trigonometric Functions
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Reference Angle
Reference Angle
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Trigonometric Functions of Any Angle
Trigonometric Functions of Any Angle
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Terminal Point
Terminal Point
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Using the Unit Circle
Using the Unit Circle
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Radius Vector
Radius Vector
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Trigonometric Functions of Angles in Other Quadrants
Trigonometric Functions of Angles in Other Quadrants
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CAST Rule
CAST Rule
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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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