Applications of Trigonometric Functions Chapter 7 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAA

Right Triangle Trigonometry; Applications Section 7.1

Trigonometric Functions of Acute Angles Right 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 Angles Non-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 Angles Example. 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 cosine tangent and cotangent secant and cosecant Theorem. [Complementary Angle Theorem] Cofunctions of complementary angles are equal

Complementary Angle Theorem Example Problem: 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 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 Elevation Angle 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 Triangles Applications of Right Triangles

The Law of Sines Section 7.2

Solving Oblique Triangles Oblique Triangle: A triangle which is not a right triangle Can have three acute angles, or Two 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 with Two angles One angle and one other side The 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 Case Information may result in One solution Two solutions No 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 Triangles Law of Sines Solving SAA Triangles Solving ASA Triangles Solving SSA Triangles Solving 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 Theorem Take ° = 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?

Key Points Law of Cosines Solving SAS Triangles Solving SSS Triangles Solving 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 Triangle Area of SAS Triangles Area 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?

Simple Harmonic Motion Example. (cont.) (c) Problem: What is the time required for one oscillation? Answer: (d) Problem: What is the frequency?

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:

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 Motion Damped Motion Graphing the Sum of Two Functions