7.1Right Triangle Trigonometry

A triangle in which one angle is a right angle is called a right triangle. The side opposite the right angle is called the hypotenuse, and the remaining two sides are called the legs of the triangle.

cb

a

90

Find the value of each of the six trigonometric functions of the angle

Adjacent

12 13c = Hypotenuse = 13

b = Opposite = 12

a

b

c

Adjacent = 5

Opposite =

Hypotenuse =

12

13

a

b

c

To solve a right triangle means to find the missing lengths of its sides and the measurements of its angles.

b35

4c

25

70

h

h = 23.49

7.2 The Law of Sines

If none of the angles of a triangle is a right angle, the triangle is called oblique.

All angles are acute

Two acute angles, one obtuse angle

To solve an oblique triangle means to find the lengths of its sides and the measurements of its angles.

CASE 1: ASA or SAA

S

A

AASA

SA A

SAA

S

SA

CASE 2: SSA

S

SA

CASE 3: SAS

S

S

S

CASE 4: SSS

The Law of Sines is used to solve triangles in which Case 1 or 2 holds. That is, the Law of Sines is used to solve SAA, ASA or SSA triangles.

Theorem Law of Sines

7.3 Law of Cosines

We use the Law of Sines to solve CASE 1 (SAA or ASA) and CASE 2 (SSA) of an oblique triangle. The Law of Cosines is used to solve CASES 3 and 4.

CASE 3: Two sides and the included angle are known (SAS).

CASE 4: Three sides are known (SSS).

Theorem Law of Cosines

Remember to give alternate form of law of cosines!

7.4 Area of a Triangle

Theorem

The area A of a triangle is

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

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

Theorem Heron’s FormulaThe area A of a triangle with sides a, b, and c is

Find the area of a triangle whose sides are 5, 8, and 11.

Additional ExamplesPage 561: 25, 27, and 29

7.5Simple Harmonic Motion;

Damped Motion; Combining Waves

Simple harmonic motion is a special kind of vibrational motion in which the acceleration a of the object is directly proportional to the negative of its displacement d from its rest position. That is, a = -kd, k > 0.

Theorem Simple Harmonic Motion

An object that moves on a coordinate axis so that its distance d from the origin at time t is given by either

The frequency f of an object in simple harmonic motion is the number of oscillations per unit of time. Thus,

Suppose an object is attached to a pendulum and is pulled a distance 7 meters from its rest position and then released. If the time for one oscillation is 4 seconds, write an equation that relates the distance d of the object from its rest position after time t (in seconds). Assume no friction.

Suppose that the distance d (in centimeters) an object travels in time t (in seconds) satisfies the equation

(a) Describe the motion of the object.

Simple harmonic

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

A = |-15| = 15 centimeters.

Suppose that the distance d (in centimeters) an object travels in time t (in seconds) satisfies the equation

d t 15 4sin

(c) What is the time required for one oscillation?

(d) What is the frequency?

Period :

frequency oscillations per second.

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.

Suppose a simple pendulum with a bob of mass 8 grams and a damping factor of 0.7 grams/second is pulled 15 centimeters to the right of its rest position and released. The period of the pendulum without the damping effect is 4 seconds.

(a) Find an equation that describes the position of the pendulum bob.

(b) Using a graphing utility, graph the function.

(c) Determine the maximum displacement of the bob after the first oscillation.

AssignmentPage 561: 10, 18, 26, 30, and 40Page 571: 16, 18, and 28