Trigonometric Functions Concept Category 3
Goals 6 basic trig functions (geometry) Special triangles Inverse trig functions (to find the angles) Unit Circle: Trig identities a b c
The Six Basic Trig functions a adjacent Cos c hypotenuse b opposite Sin c hypotenuse b opposite Tan a adjacent Sec Csc Cot 1 cos 1 sin 1 tan Sin b Tan Cos a C is always opposite of the right angle
The sides of a right -angled triangle are given special names: The hypotenuse, the opposite and the adjacent. The hypotenuse is the longest side and is always opposite the right angle. The opposite and adjacent sides refer to another angle, other than the 90 o.
I will start posting extra practice for CC1, CC, and CC3 online if you want to start preparing for the finals Suggestion: Focus on passing two CCs first
Sin Cos Opp Hyp Adj Hyp hypotenuse opposite Tan Opp Adj adjacent
Trig Functions For example evaluate sin 40 using sin key You should get: 0.64787
Some Sine Practice Function Try each of these on your calculator: sin 55 = 0.819 sin 10 = 0.174 sin 87 = 0.999
Where to use these trig functions (ratios).
Goal Problem: 34 15 cm x cm How do we solve x???
34 15 cm x cm How do we solve x??? Ask yourself: In relation to the angle, what pieces do I have? Opposite and hypotenuse What trig ratio uses Opposite and Hypotenuse? SINE Set up the equation and solve: (15) (15) sin 34 x 15 (15)Sin 34 = x 8.39 cm = x
Ex) 53 1 cm x cm Ask yourself: In relation to the angle, what pieces do I have? Ask yourself: Opposite and adjacent What trig ratio uses Opposite and adjacent? (1) Tan 53 = x (1) 1 (1)tan 53 = x 15.9 cm = x tangent Set up the equation and solve:
x cm Ask yourself: In relation to the angle, what pieces do I have? 68 18 cm Adjacent and hypotenuse Ask yourself: What trig ratio uses adjacent and hypotnuse? cosine Set up the equation and solve: (x) Cos 68 = 18 (x) x (x)cos 68 = 18 cos 68 cos 68 X = 18 cos 68 X = 48.05 cm
Ex) From a point 80m from the base of a tower, the angle of elevation is 8. How tall is the tower? x 80 8 Using the 8 angle as a reference, we know opp. and adj. sides. Use opp adj tan tan 8 = x 80 80 (tan 8 ) = x 80 (.5317) = x x 4.5 4.5 m
Ex ) A ladder that is 0 ft is leaning against the side of a building. If the angle formed between the ladder and ground is 75, how far will Coach Jarvis have to crawl to get to the front door when he falls off the ladder (assuming he falls to the base of the ladder)? 0 75 building Using the 75 angle as a reference, we know hypotenuse and adjacent side. adj Use cos cos 75 = hyp x 0 x 0 (cos 75 ) = x 0 (.588) = x x 5. 5. ft.
Ex 3. When the sun is 6 above the horizon, a building casts a shadow 18m long. How tall is the building? x 6 18 shadow Using the 6 angle as a reference, we know opposite and adjacent side. opp Use x tan tan 6 = adj 18 18 (tan 6 ) = x 18 (1.8807) = x x 33.9 33.9m
Inverse Trig Function to find the Angle Inverse Sine Function Using sin -1 (inverse sine): If 0.7315 = sin θ then sin -1 (0.7315) = θ angle
More Examples: 1. sin x = 0.1115 find angle x. x = sin -1 (0.1115) sin -1 0.1115 = x = 6.4 o. cos x = 0.8988 find angle x x = cos -1 (0.8988) cos -1 0.8988 = x = 6 o
cm 4 cm θ This time, you re looking for angle ɵ Ask yourself: In relation to the angle, what pieces do I have? Opposite and hypotenuse What trig ratio uses opposite and hypotenuse? sine Set up the equation : Sin θ = /4 Use the inverse function to find an angle Sin -1 (/4) = θ 31.59 = θ
Example C cm Find an angle that has a tangent (ratio) of /3 B 3cm A Process: I want to find an ANGLE I was given the sides (ratio) Tangent is opp/adj TAN -1 (/3) = 34 Angle A
1. H 14 cm We have been given the adjacent and hypotenuse so we use COSINE: Cos A = 6 cm A C Cos A = Cos C = h a 14 6 Cos C = 0.486 C = cos -1 (0.486) C = 64.6 o adjacent hypotenuse
. Find angle x x 3 cm A 8 cm O Tan A = Tan x = Tan x =.6667 o a 8 3 Given adj and opp need to use tan: Tan A = opposite adjacent x = tan -1 (.6667) x = 69.4 o
Do it Now: C Solve the right triangle: c =? angle B =? angle A =? B 3 c A
C Solution: 3 (hypotenuse) = (leg) + (leg) c = 3 + c = 9 + 4 c = 13 c = 13 c 3.6 B Pythagorean Theorem c A
continued Then use a calculator to find the measure of B: tan 33.7 3 1 o Then find A: ma = 180-90 - mb 56.3
Goal Problem: Space Shuttle: During its approach to Earth, the space shuttle s glide angle changes. When the shuttle s altitude is about 15.7 miles, its horizontal distance to the runway is about 59 miles. What is its glide angle? Round your answer to the nearest tenth.
Solution: You know opposite and adjacent sides. Which trig ratio (function) can you use? Glide = x tan x = opp. distance to runway adj. 59 miles Use correct ratio altitude 15.7 miles tan x = 15.7 59 Substitute values Use inverse function: Tan-1 (15.7/59) 14.9 When the space shuttle s altitude is about 15.7 miles, the glide angle is about 14.9.
Part b) When the space shuttle is 5 miles from the runway, its glide angle is about 19. Find the shuttle s altitude at this point in its descent. Round your answer to the nearest tenth. The shuttle s altitude is about 1.7 miles. Glide = 19 tan 19 = opp. tan 19 = h distance to runway adj. 5 5 tan 19 = h 5 5 miles altitude h Use correct ratio Substitute values 5 Isolate h by multiplying by 5. 1.7 h Approximate using calculator
Types of Angles The angle that your line of sight makes with a line drawn horizontally. Angle of Elevation Line of Sight Angle of Elev ation Horizontal Line Angle of Depression Horizontal Line Angle of Depression Line of Sight
Nov8 Warm-up: Do These Now Find all key features; sketch : f( x) f( x) x x x 1 x3 x x
Unit Circle Introduction Reminder: Pythagorean Theorem Angle
On an x-y plane Thus,a b c x y r
It s about a circle and a triangle. r = radius The chosen angle is always attached to the origin (0,0)
Trig Functions + xy coordinate plane You need to remember these formulas for the final
Since x y r if you think about it : x y r... Conics r r r x y 1... radius 1 r r From yesterday : Thus (cos ) (sin ) 1
Proof! o o (cos 35 ) (sin 35 )? o o (cos 5 ) (sin 5 )? o o (cos 300 ) (sin 300 )?
How about these guys? x y r x x x x y r y y y Rewrite the equations using Trig functions
Proof 1 (tan 30 0 ) (sec30 0 )??? But, your calculator doesn t have a sec key.
1 1 (tan 30 ) ( ) cos30 0 0
Unit Circle (calculator practice) Try : sin 0 o cos 0 o sin 45 o cos 45 o sin 360 cos 360 o o
Unit Circle (calculator practice ) Try : o sin 0. o cos 0. o sin 0.707 o cos 0.707 o sin 0.707 o cos 0.707
Special Triangles: Find the 6 Trig Functions (Ratios) for each
sin 30 a 1 a o 3a 3 cos 30? calculator a o a 1 tan 30? calculator 3a 3 sin 60 cos 60 tan 60 sin 45 cos 45 tan 45 o o o o o o o calculator o sin 30? o cos 30? o tan 30?
Not just for fractions
Unit Circle: circle with center at (0, 0) and radius = 1 x y r r 1 (-1,0) (0,1) (0,-1) (1,0) o sin o cos o tan y r x r y x So points on this circle must satisfy this equation.
This about this. cos sin o o x x r 1 y y r 1 x y Thus : ( x, y) (cos,sin ) o o
Handout :
Angle first 150 π 90 / 10 π / 3 3π / 4 135 5π / 6 1 0,1 1,0 180 π 0 0 1,0-1 1 7π / 6 10 5 5π / 4 7π / 4 40 4π / 3 5π 300 / 3 3π 70 / -1 0, 1 π 60 / 3 π / 4 45 π / 6 315 30 11π / 6 330
sin 0 o sin 30 o cos 0 o cos 30 o tan 0 o tan 30 o (1,0) 30 o
sin 45 o sin 60 o cos 45 o cos 60 o tan 45 o tan 60 o 45 o
sin 90 cos 90 tan 90 o o o (0,1)
r 1 and ( x, y) cos,sin 3 1, -1 1,0 3 1,, π, 150 10 1 3, 10 135 5 40 1 3, 1-1 70 0,1 1 1 1 60 45 30 1/ 0, 1 3 1 3, 3 315 300 1 3, 1 1, or, 330 1/ 0 3 1, 1, 1,0 3 1,
( x, y) ( xy, ) ( x, y) ( x, y)
(cos30 o,sin 30 o ) in 4 quadrants
(cos45 o,sin 45 o ) in 4 quadrants
Radian Measure A second way to measure angles is in radians. Definition of Radian: One radian Section is the 4.1, measure Figure of 4.5, a central Illustration angle of that intercepts arc s Arc equal Length, in length pg. to 49 the radius r of the circle. In general, s r 59
Section 4.1, Figure 4.7, Common Radian Angles, pg. 49 Radian Measure 60
Formulas: Conversions Between Degrees and Radians 180 1. To convert degrees to radians, multiply degrees by. To convert radians to degrees, multiply radians by 180 61
Use Use Change 140º to Radians 180 Change 7 3 to degrees degree to rads (radians). 140 7 140.443460953 180 180 9 180 rads (radians) to degrees 7 180 160 40 3 3
Before. 3 1, -1 1,0, 180 150 1 3, 135 10 1 90 0,1 60 1 3, 45 30, 0 3 1, 1 1,0 3 1, 10, 5 40 1 3, -1 70 0, 1 300 315 330 1 3, 3 1,,
Angle : Degree v. s. Radian 3 1, -1 1,0 3 1,, π, 5π / 6 7π / 6 1 3, 1 3, 1 π / π / 3 3π / 4 5π / 4 4π / 3-1 0,1 1 1 1 60 45 30 1/ 0, 1 3 1 3, π / 3 π / 4 3 1 3, π / 6 11π / 6 7π / 4 5π / 3 3π /, 1/ 0 3 1, 1, 1,0 3 1,
3 1, -1 1,0, π 5π / 6 1 3, 1 π / π / 3 3π / 4 0,1 1 3, π / 3 π / 4 π / 6, 0 3 1, 1 1,0 3 1,, 7π / 6 5π / 4 4π / 3 1 3, -1 7π / 4 5π / 3 3π / 0, 1 11π / 6 1 3,, 3 1,
: 0 6 3 : 5 4 4 : 0 3 6 Reference Angles Can you see a pattern???? Quadrant I II III IV 5 7 11,, 6 6 6 3 5 7,, 4 4 4 4 5,, 3 3 3
Example Find the exact value of the following: cos 3 4 Reference Angle: Cosine of Reference Angle: cos 4 4 o 3 180 Quadrant of Reference Angle: 135 4 Second Quadrant Sign of Cosine in Second Quadrant:,,,, Therefore: 3 cos 4
Do Now:
Summary of CC3 6 basic trig functions: sin, cos, tan, etc Special Triangles (exact values) How to solve for an angle; values (ratio) given a trig function Unit Circle: x y r and r 1 Unit Circle + Special Triangles + Trig Functions Radian converting to Degree (of an angle)
CC1 Final Suggestions: Do finding limits given a graph and using limits to sketch a graph first: they are the least time consuming; then sketch rational graph and finding limits given a piecewise function; then IROC or AROC
CC1 : More Final Review Problems a) Sketch a graph given the description: lim f ( x) 0 lim f ( x) 0 lim f ( x) x x x lim f ( x) f (0) 0 lim f ( x) x x4 lim f( x) x4 b) Find key features and sketch: y x 7x6 4x 7x c) Find the slope and the equation of the tangent line (AROC) for : 14 f ( x) when x 4 x 3
14 14 14 ( x 3) f (4) f ( x) f ( a) x 3 x 3 x3 ( x3) AROC x a x 4 x 4 x 4 14 x 6 x 8 ( x3) ( x3) x 8 1 ( x 4) 1 x 4 x 4 ( x 3) x 4 ( x 3) x 4 x 3 Slope lim x4 x 3 7 Equation : y mx b y x b (4) b 7 7
Practice Now No Calculator
CC3 - Section 3 Verifying Identities: 1. Solving equation? PEMDAS reverse. Factor an expression, add fractions, square a binomial, or create s monomial denominator, if possible. 3. Use the fundamental identities, whenever possible. 4. Convert all terms to sines and cosines.
csc *RECIPROCAL IDENTITIES* 1 1 sec 1 cot sin cos tan sin sin cosec tan *QUOTIENT IDENTITIES* sin cos cot cos sin *PYTHAGOREAN IDENTITIES* tan 1 sec cos 1 1cot csc EVEN-ODD IDENTITIES sin cos cos tan tan cosec sec sec cot cot
Where did our pythagorean identities come from?? Do you remember the Unit Circle? What is the equation for the unit circle? x + y = 1 What does x =? What does y =? (in terms of trig functions) cos θ + sin θ = 1 Pythagorean Identity!
Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by cos θ sin θ + cos θ = 1. cos θ cos θ cos θ tan θ + 1 = sec θ Quotient Identity another Pythagorean Identity Reciprocal Identity
Take the Pythagorean Identity and discover a new one! Hint: Try dividing everything by sin θ sin θ + cos θ = 1. sin θ sin θ sin θ 1 + cot θ = csc θ Quotient Identity a third Pythagorean Identity Reciprocal Identity
More on Pythagorean Identities sin + cos = 1 sin = 1 - cos cos = 1 - sin tan + 1 = sec tan = sec - 1 cot + 1 = csc cot = csc - 1
Guidelines for Verifying/Simplifying Identities: 1. Start with the most complicated side of the equation.. Factor an expression, add fractions, square a binomial, or create a monomial denominator, if possible. 3. Use the fundamental identities, whenever possible. 4. Convert all terms to sines and cosines. 85
Example : Verify the identity sec 1 sin. sec sec 1 (tan 1) 1 Start with the more complicated side. Pythagorean Identity sec sec tan sec tan (cos ) Simplify. Reciprocal Identity sin (cos ) cos sin Quotient Identity Simplify. 86
Graphing Utility: Verify the identity sec 1 sin. sec Table: Identical Values 1.5 1.5 Identical Graphs Graph: 0.5 0.5 87
Example: Verify the identity (tan 1)(cos 1) tan. (tan 1)(cos 1) (sec )( sin ) Pythagorea n Identity sin cos Reciprocal Identity sin cos tan Rule of Exponents Quotient Identity 88
Happy Thursday!!! Practice: One page per pair Pick at least problems from each section. Use your notes from yesterday Your Semester final will include CC1 and from Semester1
Verify sec 1 sin sec sec 1 sec tan sec tan cos 1) Choose the left side since it is more complex ) Trig Identity 3) Reciprocal Function 4) Change to sine and cosine sin cos cos 1 sin
Verify 1) Choose the right side since it is more complex 1 1 sec 1sin x 1sin 1 1sin x 1 1 sin x 1sin x1 sin x 1sin x1 sin x ) Make one fraction 1 sin x1sin 3) Simplify 4) Trig identity 5) Can t leave in denominator! sec x x x (1 sin)(1 sin x) 1 sin x cos x
More Practice: With Complex Fractions 1)Verify: 1 1 tan cos x x Solution: x 1 tan x 1 cos x 1 cos x sin cos x cos x cos x cos x )Verify: tan tan cot x cot x x x 1 cos x Solution: sin cos sin cos tan x cot x cos x sin x sin x cos x tan x cot x sin cos sin cos cos x sin x sin xcos x sin x cos x 1 cos x cos x 1 cos sin x cos x 1 x x x x x x x x x
Example) Verify the statement: sin csc cos sin Substitute using reciprocal identity sin csc cos sin sin We are done! We've shown the LHS equals the RHS 1 cos sin 1 cos sin Using the Pythagorean Identities: sin + cos = 1 thus sin = 1 - cos
Happy Wed.!!!! 1/5 th CC3 Mastery Check Practice Verify the Identities: D]
E] D]
In establishing an identity you should NOT move things from one side of the equal sign to the other. Instead substitute using identities you know and simplifying on one side or the other side or both until both sides match.
Happy Thursday CC3 Quick Check next Tuesday (no notes) Practice Quick Check today (notes ok): 1 minutes for problems 1 and only Version A v.s. Version B Critique: 10 minutes - Turn to the person behind you (with same version) It is ok to have different solution pathways! Then do problem 3 on the practice quiz and turn it in to me x (csc xcot x)
Critique Check list: Convert to same units (all sin; all cos; sin &cos) Fraction add, subt Fraction mult, div (complex fraction) Trig identities Factoring Common Factors (reverse distribution) Difference of Square Trinomial ax bx c Rationalization
Double-Angle and Half-Angle Identities
Double-Angle Formulas Formula for sine: sin x sin xcos x Formula for cosine: cosx cos x sin x 1sin x cos x 1 Formula for tangent: tan x tan x 1 tan x
Using Double Angle Formla Verify the identity. sin x cos 4x 1 sin x cos x 1 Substitute sin x 1 sin x 1 1 sin x 11
Now you try: Verify the identity
Monday 1/30 th : Use the Double Angle Formulas (and all the basic formulas) to Verify the Identities & Solve the Equations (solve for missing angles) Tomorrow + Wed: Sum and Difference Angles Formulas Thursday: Quick Check (on gradebook)