# PreCalc Chapter 6

Ch 6 Analytic Trigonometry

Ch 6.1 Verifying Trigonometric Identities

Verifying the following Identities :
cos(θ)tan(θ)=sin(θ);
cos²(θ)/1+sin(θ) = 1-sin(θ);
sin(θ)/1+cos(θ) + 1+cos(θ)/sin(θ) = 2csc(θ);
sin²(θ)/cos(θ) = sec(θ)-cos(θ);
sin4(θ)-cos4(θ)=2sin²(θ)-1

Verifying the following Identities :
sec²(x)-1/sec²(x)= sin(x);
cos(x)/ 1-sin(x) = sec(x)+tan(x);

Verifying the following Identities :
cos(θ)tan(θ)=sin(θ);
cos(x)[csc(x)tan(x)]=cot(x)+sin(x);
cos4(t)- sin4(t)/ sin²(t) = cos²(t)-1

Verifying the Identity 1-sin(t)/cos(t) + cos(t)/ 1-sin(t) = 2 sec(t)

Verifying the Identity sec(x)/1+sec(x) = 1-cos(x)/sin²(x)   http://www.youtube.com/watch?v=4QmzqwQSt8A&list=PL86281C72D802CE05&index=65&feature=plpp_video

Verifying the Identity 1+2cot²(x)+cot4(x)/ 1-cot²(x)= csc4(x)/1-cot²(x)   http://www.youtube.com/watch?v=dpFhEqdS3bU&list=PL86281C72D802CE05&index=66&feature=plpp_video

Verifying the Identity1- sin²(x)/1-cos(x)= -cos(x) ;
Verifying 1/csc(y)-cot(y)=csc(y)+cot(y) ;
Verifying cot(y)-tan(y)/sin(y)cos(y) = csc²(y)-sec²(y)

Verifying sec4(u)-sec²(u)=tan²(u)+tan4(u);
Finding a Number to Prove that sin(t+π)≠sin(t)  ;
Substituting √a²-x² /x in x=a cos(θ) and Simplify, where -π/2<θ<π/2 and a>0

Ch 6.2 Trigonometric Equations

Solving the following Equations on the Interval [0, 2π0 and then Over All Radian Solutions
2sin(θ)-1=0;
2cos(θ)+√2=0;
√3 tan(θ)-1=0;
4cos(θ)-6=cos(θ)

Solving tan²(θ)sin(θ)=tan²(θ) on Interval [0,2π)

Solving  cos(x)=-π/3;
Solving  2sin(3θ)+√2=0;
Solving  cos(4x-π/4)=√2/2;
Solving 3-tan²(β)=0

Solving 2cos(x)sin(x)=sin(x) on the Interval [0,360°)

Solving sin(x/2)=√2 – sin(x/2) by Half Angle (Part 1 of the video)

Solving 3 tan²(x)-1=0 on the Interval [0,2π)

Solving  tan(α)+tan²(α)=0;
Solving  2tan²(u)+sin(u)-6=0;
Solving sin(x)+cos(x)cot(x)=csc(x);
Solving Temperature Function T=36sin[2π/365(t-101)]+14 for t given T=-4°F

Solving tan²(θ)-1=0 on the Interval [0,2π) ;
Solving 2cos²(θ)-√3 cos(θ)=0 on the Interval [0,2π);
Solving 2sin²(θ)= -3sin(θ)-1 on the Interval [0,2π)

Solving 2cos²(x)sin(x)=1 on the Interval [0,2π)

Solving cos²(x)-cos(x)-2=0 on the Interval [0,2π)

The Function T(x)=19sin(π/6 x – π/2)+53 Modeling the Average Monthly Temperature of Water in a Mountain Stream ;
The Function S(x)=1600cos(π/6x +π/12)+1500 Modeling the Average Monthly Sales in the Month x

Ch 6.3 The Addition and Subtraction Formulas

Determining whether cos(30°+45°)=cos(45°);
Finding the Exact Value of (15°);
Finding the Exact Value of  sin(7π/12);
Finding the Exact Value of  cos(π/2 – u);
Finding the Exact Value of cos(θ- 3π/2);
Cofunction Formulas

Finding sin(α+β) and tan(α+β) if α and β are in Quadrant II and sin(α)=2/3 and cos(β)=-1/3

Finding the Exact Value of sin(7π/6 + π/4)

Finding the Exact Value of cos(15°)

Finding the Exact Value of cos(255°)

Finding the Exact Value of sin(α+β) where sin(α)=24/25 in Quadrant I and sin(β)=3/5 in Quadrant II

Find the Exact Value of tan(75°)

Ch 6.4 MultipleAngle Formulas

Does sin(60°)=sin(2•30)=2sin(30°)?
Does sin(30°)=sin(1/2 • 60°)= 1/2 sin(60°)?
Double Angle Formulas ;
Half Angle Formulas ;
Finding sin(2u),  cos(2u),  and tan(2u) if sin(u)=-4/5 where 270°<u<360°

Finding sin(u/2),  cos(u/2),  and tan(u/2) if csc(u)=-5/3 where -90°<u<0° ;
Evaluating tan(u/8);
Finding Zeros of y=cos(x)-sin(2x), for x on [0,2π)

Finding sin(2u) if sec(u)=-5/2 where π/2<u<π ;
Writing the Formula and Interval for sin(u/2)

Finding cos(u/2) if cos(u)=7 where π<u<3π/2;
Writing the Formula for cos(2u)

Finding tan(u/2) if sin(u)=5/13 where π/2<u<π ;
Writing the Formula for tan(2u)

Ch 6.5 Product–to–Sum and Sum–to–Product Formulas

Expressing sin(4x)cos(3x) as a Sum ;
Expressing sin(3x)sin(x) as a Sum ;
Expressing sin(5x)-sin(3x) as a Product ;
Solving sin(5x)+sin(x)=0 ;
Solving cos(x)-cos(3x)-sin(2x)=0 for x on [-2π,2π]

Ch 6.6 The Inverse Trigonometric Functions

Evaluating  cos-1(√2/2);
Evaluating  sin-1(-1/2);
Evaluating  arccos(-1);
Evaluating  arcsin(π/2);
Evaluating  arctan[tan(π/4)];
Evaluating  ;
Evaluating  ;
Evaluating