Trigonometric Functions Quiz

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Questions and Answers

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?

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?

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?

<p>sin t = 7/25</p> Signup and view all the answers

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?

<p>sin a = 3/5</p> Signup and view all the answers

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?

<p>cos a = 4/5</p> Signup and view all the answers

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?

<p>tan a = 3/4</p> Signup and view all the answers

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?

<p>csc a = 5/3</p> Signup and view all the answers

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?

<p>sec a = 5/4</p> Signup and view all the answers

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?

<p>cot a = 4/3</p> Signup and view all the answers

What is the exact value of sin π/3 using side lengths?

<p>sin π/3 = √3/2</p> Signup and view all the answers

If sin t = 5/12, what is the value of cos (-t)?

<p>cos (-t) = 5/12</p> Signup and view all the answers

If csc t = 2, what is the value of sec (-t)?

<p>sec (-t) = 2</p> Signup and view all the answers

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)?

<p>The length of the side opposite angle a is 7.</p> Signup and view all the answers

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)?

<p>The length of the side adjacent to angle a is 7√3.</p> Signup and view all the answers

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?

<p>The length of the side opposite the angle of π/3 is 10√3.</p> Signup and view all the answers

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?

<p>The length of the side adjacent to the angle of π/3 is 10.</p> Signup and view all the answers

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?

<p>The height of the tree is approximately 46.2 feet.</p> Signup and view all the answers

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?

<p>The height of the windowsill is approximately 52 feet.</p> Signup and view all the answers

Flashcards

Sine (sin t)

The ratio of the side opposite the angle to the hypotenuse in a right triangle.

Cosine (cos t)

The ratio of the side adjacent to the angle to the hypotenuse in a right triangle.

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)

The reciprocal of the cosine function (1/ cos t).

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Cosecant (csc t)

The reciprocal of the sine function (1/ sin t).

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Cotangent (cot t)

The reciprocal of the tangent function (1/ tan t).

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Trigonometric Functions on Unit Circle

Trigonometric functions of angles related to the unit circle. They are defined using the coordinates of a point on the circle.

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Complementary Angles

Angles that add up to 90 degrees.

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Cofunction Identities

Trigonometric functions whose values are equal for complementary angles. For example, sin(t) = cos(90-t).

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Standard Position Angle

The angle formed by the terminal side of an angle and the x-axis. Measured counterclockwise from the positive x-axis.

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Right Triangle in Standard Position

A right triangle positioned in the coordinate plane with one vertex at the origin and the right angle on the x-axis.

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Special Angles

30 degrees, 45 degrees, and 60 degrees.

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Solving Right Triangles

Finding the missing sides or angles of a right triangle using trigonometric ratios.

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Applied Problems with Right Triangles

Using trigonometric ratios to solve real-world problems involving right triangles.

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Tangent of an angle

The ratio of the opposite side to the adjacent side in a right triangle.

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Cotangent of an angle

The ratio of the adjacent side to the opposite side in a right triangle.

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Secant of an angle

The ratio of the hypotenuse to the adjacent side in a right triangle.

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Cosecant of an angle

The ratio of the hypotenuse to the opposite side in a right triangle.

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Triangle Inscribed in Unit Circle

A triangle inscribed in the unit circle, used to visualize and define trigonometric functions.

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Trigonometric Values of Special Angles

The trigonometric functions of the special angles: 30 degrees (Ï€/6), 45 degrees (Ï€/4), and 60 degrees (Ï€/3).

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30-60-90 Triangle Ratios

The hypotenuse of a right triangle with one angle of 30 degrees and one angle of 60 degrees is twice the length of the shorter leg.

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45-45-90 Triangle Ratios

The legs of a right triangle with two angles of 45 degrees are equal in length.

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Unit Circle

A circle with a radius of 1 unit.

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Exact Values of Trigonometric Functions

Finding the exact value of trigonometric functions using the ratios of side lengths in special triangles.

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Reference Angle

The angle formed by the terminal side of an angle and the positive x-axis.

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Trigonometric Functions of Any Angle

Trigonometric functions are defined for all angles in standard position, not just acute angles.

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Terminal Point

The point where the terminal side of an angle intersects the unit circle.

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Using the Unit Circle

Using the unit circle to find trigonometric values for angles.

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Radius Vector

A line segment from the origin to a point on the unit circle.

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Trigonometric Functions of Angles in Other Quadrants

Finding the values of trigonometric functions for angles beyond the first quadrant.

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CAST Rule

A rule that helps determine the signs of trigonometric functions in different quadrants.

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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 Ï€/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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