Chapter 5 â The Trigonometric Functions

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# Chapter 5 â The Trigonometric Functions - PowerPoint PPT Presentation

Chapter 5 – The Trigonometric Functions. 5.1 Angles and Their Measure. What is the Initial Side? And Terminal Side? What are radians compared to degrees? 1 radian = degrees or about 57.3 o 1 degree = radians or about 0.017 radians.

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### Chapter 5 – The Trigonometric Functions

5.1 Angles and Their Measure

What is the Initial Side? And Terminal Side?

What are radians compared to degrees?

If a is the degree measure of an angle, then all angles of the form a + 360ko, where k is an integer, are coterminal with a.

If b is the radian measure of an angle, then all angles of the form b + 2k , where k is an integer, are coterminal with b.

EX: Find one positive angle and one negative angle that are coterminal with .

EX: Identify all angles that are coterminal with a 60o angle.

Reference Angle Rule

• For any angle its reference angle is defined by
• when the terminal side is quadrant I
• when the terminal side is quadrant II
• when the terminal side is in quadrant III
• when the terminal side is in quadrant IV
• Ex: Find the measure of the reference angle for each.
• 510o
5.2 Central Angles and Arcs

A central angle of a circle is an angle whose vertex lies at the center of the circle.

Note: If two central angles in different circles are congruent then the ratio of the length of their intercepted arcs is equal to the ratio of the measures of their radii.

The length of any circular arc, s, is equal to the product of the measure of the radius of the circle, r, and the radian measure of the central angle, , that it subtends.

Find the length of an arc that subtends a central angle of 42o in a circle with radius of 8cm.

If an object moves along a circle of radius r units, then its linear velocity, v, is given by

Where is the angular velocity in radians per unit of time.

A pulley of radius 12cm turns at 7 revolutions per second. What is the linear velocity of the belt driving the pulley in meters per second?

A trucker drives 55 mph. His truck’s tires have a diameter of 26 inches. What is the angular velocity of the wheels in revolutions per second?

If is the measure of the central angle expressed in radians and r is the measure of the radius of the circle, then the area of the sector, A, is as follows:

A sector has arc length of 16cm and a central angle measuring 0.95 radians. Find the radius of the circle and the area of the sector.

5.3 Circular Functions

Def: If the terminal side of an angle in standard position intersects the unit circle at P(x,y), then and

Find each value:

For any angle in standard position with measure , a point P(x,y) on its terminal side, and , the sine and cosine functions of are as follows:

Find the values of the sine and cosine function of an angle in standard position with measure if the point with coordinates (3, 4) lies on its terminal side.

Find when and the terminal side of is in the first quadrant.

For any angle in standard position with measure , a point P(x,y) on its terminal side, and , the trigonometric functions of are as follows:

The terminal side of an angle in standard position contains the point with coordinates (8, -15). Find tangent, cotangent, secant, and cosecant for .

If and lies in quadrant III, find sine, cosine, tangent, cotangent, and secant for .

5.4 Trigonometric Functions of Special Angles

Let’s first discuss the value of each of the trig functions at

Then let’s create two very special triangles which we will then use to help find many other trig values.

5.5 Right Triangles

For an acute angle A in a right triangle ABC, the trigonometric functions are as follows:

B

c

a

A

C

b

For the following triangle find the values of the six trig functions of A.

Solve right triangle ABC. Round angle measures to the nearest degree and side measures to the nearest tenth. A = 49o

13

5

A

12

A

c

b

B

C

7

In the given triangle find the measure of angle R to the nearest degree.

• Assume that a ladder is mounted 8ft off the ground.
• How far from an 84ft burning building should the base of the ladder be placed to achieve the optimum operating angle of 60o?
• How far should the ladder be extended to reach the roof?

T

14

8

R

S

A flagpole 40ft high stands on top of the Wentworth Building. From a point P in front of Bailey’s Drugstore, the angle of elevation of the top of the pole is 54o54’, and the angle of elevation of the bottom of the pole is 47o30’. To the nearest foot, how high is the building?

5.6 The Law of Sines

Let triangle ABC be any triangle with a, b, and c representing the measures of the sides opposite the angles with measurements A, B, and C respectively. Then the following is true:

Ex: Solve triangle ABC if A = 29o10’, B = 62o20’, and c = 11.5. Round angle measures to the nearest minute and side measures to the nearest tenth.

When the measure of two sides of a triangle and the measure of the angle opposite one of them are given, there may not always be one solution. However one of the following will be true:

• No triangle exists.
• Exactly one triangle exists.
• Two triangles exist.
• Case 1: angle A less than 90o
• If a = b sin A, one solution exists
• If a < b sin A, no solution exists
• If a > b sin A, and a b one solution exists.
• If b sin A < a < b, two solutions exist.
• Case 2: angle A greater than 90o
• If a b, no solution exists.
• If a > b, one solution exists.

Ex: Solve triangle ABC if A = 63o10’, b = 18, and a = 17. Round angle measures to the nearest minute and side measures to the nearest tenth.

Ex: Solve triangle ABC if A = 43o, b = 20, and a = 11. Round angle measures to the nearest minute and side measures to the nearest tenth.

5.7 The Law of Cosines

Let triangle ABC be any triangle with a, b, and c representing the measures of sides opposite angles with measurements A, B, and C, respectively. Then, the following are true:

Ex: Solve triangle ABC if A = 52o10’, b = 6, and c = 8. Round angle measures to the nearest minute and side measures to the nearest tenth.

Ex: Solve triangle ABC if a = 21, b = 16.7, and c = 10.3. Round angle measures to the nearest minute.

5.8 Area of Triangles

We can create a new formula for area of any triangle using our understanding of the law of sines and cosines.

Find the area of triangle ABC if a = 7.5, b = 9, and C = 100o. Round your answer to the nearest tenth.

Find the area of triangle ABC if a = 18.6, A = 19o20’, and B = 63o50’. Round your answer to the nearest tenth.

Find the are of triangle ABC if a = , b = 2, and c = 3. Round your answer to the nearest tenth.

Heron’s Formula

If the measures of the sides of a triangle are a, b, and c, then the area, K, of the triangle is found as follows:

Calculate the area of triangle ABC if a = 20, b = 30, and c = 40.

If alpha is the measure of the central angle expressed in radians and the radius of the circle measures r units then the are of the segment S is as follows:

A sector has a central angle of 150o in a circle with radius of 11.5 inches. Find the area of the circular segment to the nearest tenth.