Applications of Trigonometric FunctionsChapter 7

Right Triangle Trigonometry; ApplicationsSection 7.1

Trigonometric Functions of Acute

AnglesRight triangle: Triangle in

which one angle is a right angle

Hypotenuse: Side opposite the right angle in a right triangle

Legs: Remaining two sides in a right triangle

Trigonometric Functions of Acute

AnglesNon-right angles in a right

triangle must be acute (0± < µ < 90±)

Pythagorean Theorem: a2 + b2 = c2

Trigonometric Functions of Acute

Angles

These functions will all be positive

Trigonometric Functions of Acute

AnglesExample.

Problem: Find the exact value of the six trigonometric functions of the angle µ

Answer:

Complementary Angle Theorem

Complementary angles: Two acute angles whose sum is a right angle

In a right triangle, the two acute angles are complementary

Complementary Angle Theorem

Complementary Angle Theorem

Cofunctions: sine and cosinetangent and cotangentsecant and cosecant

Theorem. [Complementary Angle Theorem]Cofunctions of complementary angles are equal

Complementary Angle Theorem

ExampleProblem: Find the exact value of

tan 12± { cot 78± without using a calculator

Answer:

Solving Right Triangles

Convention: ® is always the angle opposite side a ¯ is always the angle opposite side b Side c is the hypotenuse

Solving a right triangle: Finding the missing lengths of the sides and missing measures of the angles

Convention: Express lengths rounded to two

decimal places Express angles in degrees rounded to

one decimal place

Solving Right Triangles

We know:a2 + b2 = c2

® + ¯ = 90±

Solving Right Triangles

Example. Problem: If b = 6 and ¯ = 65±,

find a, c and ®Answer:

Solving Right Triangles

Example. Problem: If a = 8 and b = 5, find

c, ® and ¯Answer:

Applications of Right Triangles

Angle of ElevationAngle of Depression

Applications of Right Triangles

Example.Problem: The angle of elevation

of the Sun is 35.1± at the instant it casts a shadow 789 feet long of the Washington Monument. Use this information to calculate the height of the monument.

Answer:

Applications of Right Triangles

Direction or Bearing from a point O to a point P : Acute angle µ between the ray OP and the vertical line through O

Key Points

Trigonometric Functions of Acute Angles

Complementary Angle Theorem

Solving Right TrianglesApplications of Right

Triangles

The Law of Sines

Section 7.2

Solving Oblique Triangles

Oblique Triangle: A triangle which is not a right triangleCan have three acute angles,

orTwo acute angles and one

obtuse angle (an angle between 90± and 180±)

Solving Oblique Triangles

Convention: ® is always the angle opposite

side a ¯ is always the angle opposite

side b ° is always the angle opposite

side c

Solving Oblique Triangles

Solving an oblique triangle: Finding the missing lengths of the sides and missing measures of the angles

Must know one side, together withTwo anglesOne angle and one other sideThe other two sides

Solving Oblique Triangles

Known information:One side and two angles: (ASA,

SAA)Two sides and angle opposite

one of them: (SSA)Two sides and the included

angle (SAS)All three sides (SSS)

Law of Sines Theorem. [Law of Sines]

For a triangle with sides a, b, c and opposite angles ®, ¯, °, respectively

Law of Sines can be used to solve ASA, SAA and SSA triangles

Use the fact that ® + ¯ + ° = 180±

Solving SAA Triangles

Example. Problem: If b = 13, ® = 65±, and

¯ = 35±, find a, c and °Answer:

Solving ASA Triangles

Example. Problem: If c = 2, ® = 68±, and ¯

= 40±, find a, b and °Answer:

Solving SSA Triangles

Ambiguous CaseInformation may result in

One solutionTwo solutionsNo solutions

Solving SSA Triangles

Example. Problem: If a = 7, b = 9 and ¯ =

49±, find c, ® and °Answer:

Solving SSA Triangles

Example. Problem: If a = 5, b = 4 and ¯ =

80±, find c, ® and °Answer:

Solving SSA Triangles

Example. Problem: If a = 17, b = 14 and ¯

= 25±, find c, ® and °Answer:

Solving Applied Problems

Example.Problem: An airplane is sighted

at the same time by two ground observers who are 5 miles apart and both directly west of the airplane. They report the angles of elevation as 12± and 22±. How high is the airplane?

Solution:

Key Points

Solving Oblique TrianglesLaw of SinesSolving SAA TrianglesSolving ASA TrianglesSolving SSA TrianglesSolving Applied Problems

The Law of Cosines

Section 7.3

Law of Cosines

Theorem. [Law of Cosines]For a triangle with sides a, b, c and opposite angles ®, ¯, °, respectively

Law of Cosines can be used to solve SAS and SSS triangles

Law of Cosines

Theorem. [Law of Cosines - Restated]The square of one side of a triangle equals the sum of the squares of the two other sides minus twice their product times the cosine of the included angle.

The Law of Cosines generalizes the Pythagorean TheoremTake ° = 90±

Solving SAS Triangles

Example. Problem: If a = 5, c = 9, and ¯ =

25±, find b, ® and °Answer:

Solving SSS Triangles

Example. Problem: If a = 7, b = 4, and c =

8, find ®, ¯ and °Answer:

Solving Applied Problems

Example. In flying the 98 miles from Stevens Point to Madison, a student pilot sets a heading that is 11± off course and maintains an average speed of 116 miles per hour. After 15 minutes, the instructor notices the course error and tells the student to correct the heading. (a) Problem: Through what angle will the

plane move to correct the heading?Answer:

(b) Problem: How many miles away is Madison when the plane turns?

Answer:

Key Points

Law of CosinesSolving SAS TrianglesSolving SSS TrianglesSolving Applied Problems

Area of a Triangle

Section 7.4

Area of a Triangle

Theorem.The area A of a triangle is

where b is the base and h is an altitude drawn to that base

Area of SAS Triangles

If we know two sides a and b and the included angle °, then

Also,

Theorem.The area A of a triangle equals one-half the product of two of its sides times the sine of their included angle.

Area of SAS Triangles

Example.Problem: Find the area A of the

triangle for which a = 12, b = 15 and ° = 52±

Solution:

Area of SSS Triangles

Theorem. [Heron’s Formula]The area A of a triangle with sides a, b and c is

where

Area of SSS Triangles

Example.Problem: Find the area A of the

triangle for which a = 8, b = 6 and c = 5

Solution:

Key Points

Area of a TriangleArea of SAS TrianglesArea of SSS Triangles

Simple Harmonic Motion; Damped Motion; Combining Waves Section 7.5

Simple Harmonic Motion

Equilibrium (rest) position

Amplitude: Distance from rest position to greatest displacement

Period: Length of time to complete one vibration

Simple Harmonic Motion

Simple harmonic motion: Vibrational motion in which acceleration a of the object is directly proportional to the negative of its displacement d from its rest position

a = {kd, k > 0 Assumes no friction or other

resistance

Simple Harmonic Motion

Simple harmonic motion is related to circular motion

Simple Harmonic Motion

Theorem. [Simple Harmonic Motion]An object that moves on a coordinate axis so that the distance d from its rest position at time t is given by either

d = a cos(!t) or d = a sin(!t)where a and ! > 0 are constants, moves with simple harmonic motion.The motion has amplitude jaj and period

Simple Harmonic Motion

Frequency of an object in simple harmonic motion: Number of oscillations per unit time

Frequency f is reciprocal of period

Simple Harmonic Motion

Example. Suppose that an object attached to a coiled spring is pulled down a distance of 6 inches from its rest position and then released.Problem: If the time for one oscillation

is 4 seconds, write an equation that relates the displacement d of the object from its rest position after time t (in seconds). Assume no friction.

Answer:

Simple Harmonic Motion

Example. Suppose that the displacement d (in feet) of an object at time t (in seconds) satisfies the equation

d = 6 sin(3t)(a) Problem: Describe the motion of

the object.Answer:

(b) Problem: What is the maximum displacement from its resting position?

Answer:

Simple Harmonic Motion

Example. (cont.)

(c) Problem: What is the time

required for one oscillation?

Answer:

(d) Problem: What is the

frequency?

Answer:

Damped Motion

Most physical systems experience friction or other resistance

Damped Motion

Theorem. [Damped Motion]

The displacement d of an

oscillating object from its at-rest

position at time t is given by

where b is a damping factor

(damping coefficient) and m is the

mass of the oscillating object.

Damped Motion

Here jaj is the displacement at

t = 0 and is the period

under simple harmonic motion

(no damping).

Damped Motion Example. A simple pendulum

with a bob of mass 15 grams and a damping factor of 0.7 grams per second is pulled 11 centimeters from its at-rest position and then released. The period of the pendulum without the damping effect is 3 seconds. Problem: Find an equation that

describes the position of the pendulum bob.

Answer:

2

32

22

-6

-4

-2

2

4

6

Graphing the Sum of Two Functions

Example. f(x) = x + cos(2x)Problem: Use the method of

adding y-coordinates to graph y = f(x)

Answer:

Key Points

Simple Harmonic MotionDamped MotionGraphing the Sum of Two

Functions