Test 3

percent decrease

( (original amount - new amount) ÷ original amount) × 100%

Difference of Squares Formula

(a+b)(a+b)= a²+2ab+b² (a-b)(a-b)= a²-2ab+b² (a+b)(a-b) = a^2 - b^2

things to know when solving inverse functions

- when problem is written like sin-1(-√2/2) or Arcsin(√3/2) KEEP IN MIND INVERSE RANGES (RESTRICTED DOMAINS) but when it's written like sinθ, tanθ, cosθ (like normal) you don't need to BECAUSE sinθ=1/2 has unlimited answers while, sin-¹(1/2) only has one -WATCH FOR QUADRANTS AND SIGNS -Watch for proper calc mode - you can "cancel" out inverse and original function if and only if whats in the (__) corresponds to the inverse functions restricted domain. WHEN THEY ARE "NICE" numbers that are on the unit circle. You know that you can solve when you plug into calc and get an answer with no decimals) ex. cos-¹(cos 89) CAN CANCEL while tan-¹(5π/4) CANNOT. (you can always solve these problems by just plugging into calc -when you have opposite functions (aka cot, sec, csc) of tan, cos, and sin; just write the corresponding inverse of the original trig function and put what's in the ( ) under 1 (aka take reciprocal) ex. sec5 -----> cos-¹(1/5) cot(-3/4)-----> tan-¹(-4/3)

Cotangent Pythagorean Identity

1 + cot² x = csc² x cot² x = csc² x - 1 1 = csc² x - cot² x

Tangent Pythagorean Identity

1 + tan² x = sec² x tan² x = sec² x - 1 1 = sec² x - tan² x

how to find area of a part of a circle with a triangle in it

1) find the area of the sector of the circle with the formula A= (1/2)(r²)(θ) 2) find the area of the triangle by using the formula for area of a triangle 3) subtract the ares of the sector of the circle for the area of the triangle. The difference is the area of the segment

to find the exact value of an expression...

1)Use sum or difference identities: make sure they are perfect angles such as 30,45,60. or 90 to make up the number.ex) Sin(165)= (120+45) 120 is a nice number because 60 + 60 =120. also will occur in radians and not only degrees When given problems like "find exact value of cos *(a-b)* given that cos a= -2/3 and sin b= 3/5 YOU MUST DRAW TWO TRIANGLES. ONE FOR A AND ONE FOR B and use pythag theorem to solve for missing parts... Watch for quadrant and signs. NOTE: you're not giving actual angle measures only side length measures. Even if the problem looks like cos[sin-¹(-5/13)+tan-¹(4/3)] still write two triangles and find missing side length ratios. Use the ratios NOT ANGLES to solve to exact value

Area of a sector of a circle

A= (1/2)(r²)(θ) r is radius and θ is in RADIANS

area of a triangle

A=(1/2)(a)(b) sinC A=(1/2)(b)(c) sinA A=(1/2)(c)(a) sinB when all sides are known use Herons formula: A=√s(s-a)(s-b)(s-c), where s= (a+b+c)/2 keep in mind the EVERYTHING is under the square root in the formula above UPPERCASE IS ANGLE; LOWERCASE IS SIDE the angle MUST by inbetween 2 sides

Double angle formula for Cosine

Cos(2θ) = cos² θ - sin²θ = 1 - 2sin²θ = 2cos²θ - 1

Sum formula for Cos

Cos(u + v) = cos u cos v - sin u sin v

Difference formula for Cos

Cos(u - v) = cos u cos v + sin u sin v

when doing test..

MAKE SURE EVERY SINGLE PROBLEM IS DONE IN THE CORRECT CALC MODE. watch for domain. When restrictions/intervals are given in radians answer must be in RADIANS. When in degrees, answer must be in DEGREES watch for quadrant and signs when solving using double angle or half-angle identities, look at given information, draw the triangle, solve for missing, and watch Q's when doing sum and difference formula watch for quadrants inverses solve for ANGLES and ratios solve for SIDE LENGTHS ALWAYS LOOK FOR SSA CASE

Double angle formula for Sine

Sin(2θ) = 2sinθcos θ

Sum Formula for Sin

Sin(u + v) = sin u cos v + cos u sin v

Difference formula for Sine

Sin(u - v) = sin u cos v - cos u sin v

SOHCAHTOA

Sin=opposite/hypotenuse Cosine=adjacent/hypotenuse Tan=opposite/adjacent ONLY USED FOR RIGHT TRIANGLES! tan, sin, cos used for side lengths and tan-¹, sin-¹, cos-¹ used for angles

Double angle formula for Tan

Tan(2θ) = 2tanθ/ 1 - tan²θ

Sum formula for Tan

Tan(u + v) = (tan u + tan v)/1 - tan u tan v

Difference formula for Tan

Tan(u - v) = (tan u - tan v)/1 + tan u tan v

Power reducer formula for Tan

Tan²θ = 1 - cos(2θ)/ (1 + cos(2θ))

Law of Cosines

To solve for sides: a²=b²+c²-2(b)(c) CosA b²= a²+c²-2(a)(c) CosB c²=a²+b²-2(a)(b) CosC can be written like this to solve for angles: cos(A) = (b^2 + c^2 − a^2)/(2bc) cos(B) = (c^2 + a^2 − b^2)/(2ca) cos(C) = (a^2 + b^2 − c^2)/(2ab) USED FOR NON-RIGHT TRIANGLES NO SSA CASE requires 3/4 of ( 1 angle and 3 sides)

finding the exact values of inverse functions

When given problems like "find exact value of sin(tan-¹(-3/4))YOU MUST DRAW A TRIANGLE FOR WHATS IN (___) use pythag theorem to solve for missing parts and then find what the value for outside of the (__) is. *THESE ARE NOT "NICE" NUMBERS. You MUST solve by using triangle to get exact values. You can check if these numbers are "nice" or not by plugging into calc* Watch for quadrant(determined by inverse domain), signs, AND DOMAIN NOTE: you're not giving actual angle measures only side length measures. Use the ratios NOT ANGLES to solve to exact value

Law of Sines

a/sinA = b/sinB = c/sinC where lowercase letters are side lengths and Upper case are angle measurements AMBIGUOUS CASE (SSA) HAPPENS ONLY WITH L.O.S USED FOR NON-RIGHT TRIANGLES requires 3/4 of (2 angles 2 sides)

relative maximums

basically asking for period if it wants two CONSECUTIVE relative maximums. (how long it takes from the beginning of one hump to the end of the second hump) when the problem says the first to the second relative maximums it means from the highest (or lowest) point on one hump to the next hightest (or lowest) point on another hump. This skips a "hump" and is basically adding another period to the graph

Power reducer formula for Cos

cos²θ = 1 + cos(2θ)/2

Waves crest

divide period by 4 (maybe) depending on sin or cos

writing algebraic expressions as functions of theta

ex) √(x²+16)/x and x=4 tanθ 1) substitute and simp √(4 tanθ²+16)/4 tanθ-->√(16 tanθ²+16)/4tanθ 2)Pull out common # √(16 (tanθ²+1))/4tanθ 3) Pull # out of radical and simp 4)√((tanθ²+1))/4tanθ --> secθ/tanθ 5) Watch for quadrant and signs (+/-) that are given in problem be careful when you pull out common factor... there may be a hidden phythag identity waiting it be converted

Harmonic motion

s(t)= acos wt s(t)= asin wt Period: Time for one complete oscillation. needs to be in ONE revolution per seconds,mins, hours etc P= 2pi/w T= 2pi/w P=1/f (seconds/cycle) B/W: Radians/unit or time (sec,min,hour). Needs to be in ONE sec,min,hour per # or revolutions b=2pi/p or take P in cycles per second and multiply by 2pi because 1 cycle= 2pi rad Frequency: The number of oscillations produced per second. Needs to be in CYCLES per ONE second. f= 1/p f=1/t f=w/2pi (cycles/second)

Sine Pythagorean identity

sin² x + cos² x = 1 cos² x = 1 - sin² x sin² x = 1 - cos² x

Power reducer formula for Sin

sin²θ = 1 - cos(2θ)/2

right triangle trig formulas for certain problems

speed= distance/time distance= speed × time time= speed/distance angular speed: ω=θ/t where θ is in RADIANS arc length: S=θr where θ is in RADIANS Linear speed: Speed=Distance/time v=s/t v=rθ/t v=rws=rθ C=2πr S=rwt

things to note when solving trigonometric equations

tips on how to solve: - ALWAYS check for extraneous solutions when you square an equation - look for u substitution, factoring, squaring both sides, pulling out CF or negative, and canceling out fractions or dividing by common factor. -when necessary, substitute components for identities FIRST, then simplify - when you see problems with squares on two different functions (like sin²θ and cos²θ) then that means you can 1) pull out a neg to make it look like a phythag ident or double angle ident OR 2) FACTOR IT - when you get stuck, ALWAYS LOOK FOR POSSIBLE IDENTITIES - you MUST make whats inside of the function match with the rest of the equation. ex. cos4x-3cos2x=1 where 4x can turn to 2x by using DOUBLE angle identity (2x is double of 4x) - when you see (sec, csc, cot) just turn to (cos, sec, tan) and take reciprocal of what's inside WATCH SIGNS (+/-)

verifying trigonometric identities

tools: - change to sins or cos -Simplify (common denom, factor, Common factor, squares and difference of squares - look for phythag identities - rewrite (-x) depending on function. AKA pulling out negative - conjugate (when you see things like 1-sec or 1+sec) - u-substitution POWER REDUCERS: when you see (1-sec^2x) you can write it out at (1-secx)(1+secx) - When you see a lot of powers, keep separating them. Watch for Pythag Idents -when you divide inverses, they give you answers that are squared. Inverses multiplied cancel out - when signs (+/-) don't match, only substitute a portion watching out for phythag idents

Half angle formula for Tan

±√(1-cosθ)/(1+cosθ) EVERYTHING UNDER ROOT