# Trigonometric Functions

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Trigonometric Functions
Chapter 5 Trigonometric Functions

Section 5.1 Angles and Degree Measure
Learn how to convert decimal degree measures to degrees, minutes, and seconds. Find the number of degrees in a given number of rotations Identify angles that are coterminal with a given angle

5.1 Angles and Degree Measure
Vertex Endpoint of an angle Initial Side The ray of the angle that is fixed Terminal Side The second ray that rotates to form the angle Standard Position An angle with its vertex at the origin and its initial side along the positive x-axis

5.1 Angles and Degree Measure

5.1 Angles and Degree Measure
Most common unit to measure an angle Minutes A degree is subdivided into 60 equal parts called minutes (1’) Seconds A minute is subdivided into 60 equal parts called seconds (1”) Example 1: Change degrees to degrees, minutes, and seconds. 15.735o=15o+(0.735*60)’ =15o+44.1’ =15o+44’+(0.1*60)” =15o+44’+6”

5.1 Angles and Degree Measure
Example 2: Change degrees to degrees, minutes, and seconds. 329o+7’+30” Example 3: Change 39o+5’+34” to degrees 39o+5’+34”=39o+5’*(1 o/60’) +34” (1 o/3600’) 39.093o Example 4: Change 35o+12’+7” to degrees 35.202o

5.1 Angles and Degree Measure
Quadrantal Angle If the terminal side of an angle that is in standard position coincides with one of the axes How many quadrantal angles are there? 4 What are their measures? 90 degrees 180 degrees 270 degrees 360 degrees

5.1 Angles and Degree Measure
Give the angle measure represented by the rotation about the axis 5.5 Rotations clockwise Which way do we rotate to go clockwise? Negative, clockwise rotations ALWAYS have negative measures 5.5 * -360 = degrees 3.3 Rotations counterclockwise Which way do we rotate to go counterclockwise? Positive, counterclockwise rotations ALWAYS have positive measures 3.3*360 = 1188 degrees 9.5 Rotations clockwise -3420 degrees 6.75 Rotations counterclockwise 2430 degrees

5.1 Angles and Degree Measure
Coterminal Angles If α is the degree measure of an angle, then all angles measuring α+ 360k degrees, where k is an integer; are coterminal with α. Two angles in standard position that have the same initial side Identify all angles that are coterminal with a 45 degree angle. Find one positive and one negative angle that are also coterminal. All angles having a measure of 45o+ 360ko Positive Angle 45o+ 360*(1)o=405o Negative Angle 45o+ 360*(-2)o=-675o

5.1 Angles and Degree Measure
Identify all angles that are coterminal with a 294 degree angle. Find one positive and one negative angle that are also coterminal. All angles having a measure of 294o+ 360ko Positive Angle 294o+ 360*(1)o=654o Negative Angle 294o+ 360*(-1)o=-66o

5.1 Angles and Degree Measure
If an angle with 775 degrees is in standard position, determine a coterminal angle that is between 0 and 360 degrees. State the quadrant in which the terminal side lies. Find the number of rotations about the axis by dividing 775 by 360 = Since we need an angle between 0 and 360 degrees, what rotation number should we use? Subtract 2 and use α= *360 Could also take the number and continue to subtract 360 until you get a number between 0 and 360. 55o What quadrant does the terminal side fall in? Quadrant 1

5.1 Angles and Degree Measure
If an angle with -777 degrees is in standard position, determine a coterminal angle that is between 0 and 360 degrees. State the quadrant in which the terminal side lies. 303o What quadrant does the terminal side fall in? Quadrant 4

5.1 Angles and Degree Measure
Reference Angle The acute angle formed by the terminal side of the given angle and the x-axis Reference Angle Rule For any angle α, 0o< α <360o, its reference angle α’ is defined by: α, when the terminal side is in Quadrant I. 180o- α, when the terminal side is in Quadrant II. α – 180o, when the terminal side is in Quadrant III. 360o- α, when the terminal side is in Quadrant IV.

5.1 Angles and Degree Measure
Find the measure of the reference angle for an angle with a measurement of 120 degrees What Quadrant does this angle’s terminal side fall in? Between 90o and 180o Quadrant II We use which formula? 180o- α 180o-120o=60o Find the measure of the reference angle for an angle with a measurement of -135degrees First we have to find a positive coterminal angle 360o-135o = 225o What quadrant does this angle’s terminal side fall in? Between 180o and 270o Quadrant III α – 180o 225o-180o=45o

5.1 Angles and Degree Measure
Find the measure of the reference angle for an angle with a measurement of 312 degrees 48o Find the measure of the reference angle for an angle with a measurement of -195 degrees 15o

Section 5.2 Trigonometric Ratios in Right Triangles
Learn how to find the values of trigonometric ratios for actue angles of right triangles

5.2 Trigonometric Ratios in Right Triangles
What is a Right Triangle? A triangle with a 90 degree angle in it. How can we classify the other two angles in the right triangle? Must be acute Are also complementary What are the parts of a Right Triangle? < 1 + < 2 = 90o 1 Side C is the hypotenuse Sides A and B are the legs 2

5.2 Trigonometric Ratios in Right Triangles
When looking at one specific acute angle in a triangle, we can classify the legs by: Adjacent Side: the leg that is a side of the acute angle Opposite Side: the leg that is the side opposite the angle Looking at Triangle ABC, what are the adjacent and opposite sides for < B? B Hypotenuse Adjacent Opposite

5.2 Trigonometric Ratios in Right Triangles
The ratios of the sides of right triangles based on a specific acute angle within the right triangle Easy way to remember Sin, Cosine, and Tangent Ratios SOHCAHTOA Sine Ratio of the side opposite Θ and the hypotenuse Cosine Ratio of the side adjacent Θ and the hypotenuse Tangent Ratio of the side opposite Θ and the side adjacent to Θ

5.2 Trigonometric Ratios in Right Triangles

5.2 Trigonometric Ratios in Right Triangles
Find the values of sine, cosine, and tangent for <B. What do we need to do first? Find the third side’s length How do we do that? Pythagorean Theorem, A2 + B2 = C2 C=3*(85)1/2 Which angle is our Θ? Sin Θ Opposite/hypotenuse 18/33 Cos Θ Adjacent/hypotenuse 3*(85)1/2/33 Tan Θ Opposite/Adjacent 18/3*(85)1/2 18 m A C 3*(85)1/2 m 33 m Θ B

5.2 Trigonometric Ratios in Right Triangles
Find the values of sine, cosine, and tangent for <A. Find the third side’s length C=17 m Which angle is our Θ? Sin Θ 8/17 Cos Θ 15/17 Tan Θ 8/15 C 17 m 8 m Θ A B 15 m

5.2 Trigonometric Ratios in Right Triangles
Cosecant Opposite of sine, cscΘ = 1/sin Θ Hypotenuse/side opposite Secant Opposite of cos, secΘ = 1/cosΘ Hypotenuse/side adjacent Cotangent Opposite of tan, cotΘ = 1/tanΘ Side Adjacent/ Side opposite

5.2 Trigonometric Ratios in Right Triangles
If cos Θ = ¾, what is sec Θ? sec Θ = 1/cos Θ =1/(3/4) =4/3 If sin Θ=0.8, what is csc Θ? =1.25 If csc Θ = find sin Θ? sin Θ = 1/csc Θ =1/1.345 =.7435 If cot Θ = 6/5, what is tan Θ? =5/6

5.2 Trigonometric Ratios in Right Triangles
Find the values of the six trigonometric ratios for <E. First find third side using pythagorean theorem (58)1/2 m Cos E = 3* (58)1/2 /58 Sin E= 7* (58)1/2 /58 Tan E=7/3 Sec E= (58)1/2 /3 Csc E= (58)1/2 /7 Cot E=3/7 D (58)1/2 m 7 m Θ E F 3 m

5.2 Continued Special Triangles
30O-60O-90O 45O-45O-90O What are the special relationships we know about these triangles?

5.2 Continued What are the special relationships we know about these triangles? 45O-45O-90O

5.2 Continued Trigonometric Ratios for 30o, 60o, 90o

5.2 Continued Cofunctions
Trigonometric functions that are equal when their arguments are complementary angles, such as sine and cosine, tangent and cotangent, and secant and cosecant. Sin Θ = Cos (90O- Θ) Cos Θ = Sin (90O- Θ) Tan Θ=Cot (90O- Θ) Cot Θ=Tan (90O- Θ) Sec Θ=Csc (90O- Θ) Csc Θ=Sec (90O- Θ)

Section 5.3 Trigonometric Functions on the Unit Circle
Find the values of the six trigonometric functions using the unit circle Find the values of the six trigonometric functions of an angle in standard position given a point on its terminal side

Trigonometric Functions on the Unit Circle
A circle with a radius of 1 Usually with the center on the origin on the coordinate system Symmetric with respect to the x-axis, y-axis, and the origin

Trigonometric Functions on the Unit Circle
Consider an angle between 0O and 90O in standard position Let P(x,y) be where the angle intersects with the unit circle Draw a perpendicular segment from intersection point back down to the positive x axis Creates a right triangle Find the sin Θ and cos Θ Sin Θ = y; Cos Θ = x

Trigonometric Functions on the Unit Circle
Sine and Cosine on the Unit Circle If the terminal side of an angle Θ in standard position intersects the unit circle at P(x,y), then cos Θ = x and sin Θ=y.

Trigonometric Functions on the Unit Circle

Trigonometric Functions on the Unit Circle
Circular Functions Functions defined using the unit circle Ie Sin and Cosine How can we define the other cosine functions on the unit circle? Tan Θ = y/x Csc Θ =1/y Sec Θ =1/x Cot Θ = x/y

Trigonometric Functions on the Unit Circle
Use the unit circle to find each value Cos (-180O) Which way do you go for negative angles? Clockwise What is the ordered pair of the intersection of this angle on the unit circle? (-1,0) Cos Θ = x-axis Cos (-180O) = -1 Sec(90O) Where is the angle located ? Terminal Side is on positive y axis Where is the intersection on the unit circle? Intersection at (0,1) Sec Θ = 1/x Sec(90O)=1/0 = undefined

Trigonometric Functions on the Unit Circle
Use the unit circle to find each value Sin(-90O) -1 Cot(270O)

Trigonometric Functions on the Unit Circle
Use the unit circle to find the values of all six trigonometric functions for a 210 degree angle. What is the intersection with the unit circle? (-√(3) /2, -1/2) Sin Θ = -1/2 Cos Θ =- √(3) /2 Tan Θ = √(3) /3 Csc Θ =-2 Sec Θ =-2 √(3) /3 Cot Θ = √(3)

Unit Circle Quiz Use the unit circle to find each value
Tan 3600 Cos 450 Sin(-600) Csc(-2100 ) Sec(2250 ) Name the six trigonometry functions for the angles below 1500 4200

Trigonometric Functions on the Unit Circle
What if the angle doesn’t fall within the unit circle? What if the length is greater or less than 1? Use length R instead of 1 in unit circle and R = (x2+y2)1/2

Trigonometric Functions on the Unit Circle
What are the six trigonometric values using length r? Sin Θ =y/r Cos Θ =x/r Tan Θ =y/x Csc Θ =r/y Sec Θ =r/x Cot Θ =x/y

Trigonometric Functions on the Unit Circle
Find the six trigonometric functions for angle Θ in standard position if a point with the coordinates (-15, 20) lies on its terminal side. Draw Figure on xy axis What are the side measures? Leg = -3 Leg = 4 Hypotenuse = 5 Where is the theta located? At the point they gave us What are the six functions? Sin Θ =4/5 Cos Θ =-3/5 Tan Θ=-4/3 Csc Θ =5/4 Sec Θ =-5/3 Cot Θ =-3/4

Review Quiz 5.3 Suppose Θ is an angle in standard position whose terminal side lies in Quadrant III. If sin Θ = -4/5, find the values of the remaining five trigonometric functions of Θ? What do we need to do first? Draw figure Next? Find Missing Side +/-3, use negative 3 because Quadrant III Find 5 remaining trig functions. Cos Θ = -3/5 Tan Θ =4/3 Csc Θ =-5/4 Sec Θ =-5/3 Cot Θ =3/4

5.4 Applying Trigonometric Functions
Use trigonometry to find the measures of the sides of right triangles Make sure calculators are in degrees and not radians Press Mode Scroll down to third line and arrow left one Hit enter Quit Quick Check Cos(90) = 0

Applying Trigonometric Functions
In triangle PRQ, P = 35o and r = 14. Find Q. First draw figure How is Q related to Θ? Adjacent Side What function should we use? Cosine Cos P = q/r Cos 35=q/14 14*cos(35)=Q Q is about 11.5

Applying Trigonometric Functions
Angles of Elevation The angle between a horizontal line and the line of sight from an observer to an object at a higher level Angles of Depression The angle between a horizontal line and thel ine of sight from the observer to an object at a lower level

5.5 Solving Right Triangles
Evaluate inverse trig functions. Find missing angle measurements. Solve right triangles.

Solving Right Triangles
Inverse of a Trigonometric Function The arcsine, arccosine, and arctangent relations with their corresponding trigonometric functions Arcsine Sin x = √3/2 Can be written as x = arcsine √3/2 or x = sin-1 √3/2 Read this as x is an angle whose sine is √3/2 Same for other two trig functions Arccosine Arctangent

Solving Right Triangles
Sin x = √3/2 X is an angle with sine √3/2 X = arcsine √3/2 or x = sin-1 √3/2 60 o, 120 o, or any coterminal angles with these Tan x = 1 45 o,225o Sin x = -1/2 210 o, 330 o

Solving Right Triangles
Evaluate and assume angles are in Quadrant I Cos(arcsin 2/3) Let B = arcsin 2/3 Sin B = 2/3 Draw figure in Quadrant I Find X = √5/3 Try in calculator Tan(cos-1 4/5) 3/4

Solving Right Triangles
Solve the Triangle with A = 33o and B = 5.8 First fill in known amounts Find Angle B Two acute angles in a right triangle are complementary B =90-33o = 57o degrees Find side A Tan A = a/b Tan 33o = a / 5.8 5.8 Tan 33o = a a = 3.767 Find side C Cos A = b/c Cos 33o = 5.8/c C=5.8/cos 33o c=6.916 B A C

Solving Right Triangles
Solve the Triangle with K = 40o and k = 26 < L = 50o l = 31.0 j = 40.4 K J L

Section 5.6 The Law of Sines
Solve triangles by using the Law of Sines if the measures of two angles and one side are given Find the area of a triangle if the measures of sides and the included angle or the measures of two angles and a side are given.

The Law of Sines Let Triangle ABC by any triangle with a, b, and c representing the measures of the sides opposite the angles with measures A, B, and C, respectively. Then the following is true: a = b = c SinA SinB SinC

The Law of Sines Solve triangle ABC if A = 33o, B = 105o, and b = 37.9
Find < C by subtracting the other two angle measures from 180 C= 42o a = Sin33o Sin105o A = 21.37o c = Sin42o Sin105o C=26.25o

The Law of Sines Area of a Triangle
Let triangle ABC be any triangle with a, b, and c representing the measures of the sides opposite the angles with measurements A, B, C, respectively. Then the area K can be determined using: K = ½ * b* c* sinA K = ½ * a* c* sinB K = ½ * b* a* sinC

The Law of Sines Find the area of triangle ABC if a = 4.7, c = 12.4, and B = 47o 20’ First convert B to decimal degrees B = 47.33o K = ½ * a* c* sinB K = ½ * 4.7*12.4*sin47.33o K = 21.4 units2 Find the area of triangle ABC if b = 21.2, c = 16.5, and A = 25o K = ½ * b* c* sinA K = 73.9 units2

The Law of Sines Area of a Triangle
Let triangle ABC be any triangle with a, b, and c representing the measures of the sides opposite the angles with measurements A, B, C, respectively. Then the area K can be determined using: K = ½*a2*sinB*sinC SinA K = ½*b2*sinA*sinC SinB K = ½*c2*sinB*sinA SinC

The Law of Sines Find the area of triangle DEF if d = 13.9, D = 34.4o, and E = 14.8o Find the measure of <F first, <F = 130.8o K = ½*d2*sinE*sinF SinD K = ½*13.92*sin 14.8o *sin 130.8o SinD 34.4o K = units2

Section 5.8 The Law of Cosines
Let Triangle ABC by any triangle with a, b, and c representing the measures of the sides opposite the angles with measures A, B, and C, respectively. Then the following is true: a2=b2 + c2 -2*b*c*cosA b2=a2 + c2 -2*a*c*cosB c2=b2 + a2 -2*b*a*cosC

The Law of Cosines Solve the triangle with the following information:
A = 120o,b=9, c=5 Solve for Side a a2=b2 + c2 -2*b*c*cosA a2= – 2*9*5*cos(120o) a2=426.97 a=12.3 Now use Law of Sines to find <B = 9 Sin120o sinB B=arcsin (9*sin120o) 12.3 B= 39.3o C= 180o – 120o-39.3o C=20.7o

The Law of Cosines Try a triangle with A = 39.4o,b = 12, c = 14
B=58.2o C = 82.4o a = 9.0 Try a triangle with a =19, b = 24.3, c = 21.8 A = 48.3o B = 72.7o C = 59o

The Law of Cosines Hero’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: K = √s*(s-a)*(s-b)*(s-c) Where S = ½ (a+b+c)

The Law of Cosines Find the area of the triangle with the following information: a=72 cm b=83 cm c=95 cm K = √s*(s-a)*(s-b)*(s-c) Where S = ½ (a+b+c) S=125 K= cm2