Graphing Sine & Cosine
Quadrant Review
Review:
1. Identify whether sin, cos, and
tan are positive or negative in
each quadrant.
2. Find the values of
sin, cos, and tan at
θ=π/2, π, 3π/2, & 2π.
Find two values of θ that satisfy the equation.
cos θ = 1/2
cos θ = -1/2
tan θ = √3 tan θ = -√3
cot θ = 0 tan θ = 0
Solving Trig Equations for the Complete Solution
1. _________________ the Trig function.
2. Draw possible ______________________.
3. Use _________________________ angle.
4. Complete solution includes __________________________ angles.
Solve 2sinx -1 = 0 Solve cos x + √2 = -cos x
Solve tan2 x - 3 = 0 Solve csc x -2 = 0
Factoring To Solve
Find all solutions of 2sin2 x - sin x - 1 = 0
Find all solutions of sec4 x - 4sec2 x = 0
Solving Functions with Multiple Angles & Solutions on an Interval
Solve 3tan [pic]+ 3 = 0
Solve 2cos 3t – 1 = 0
Applications of Solving Trig Functions
The monthly sales (in thousands of units) of a seasonal product are approximated by
S = 74.50 + 43.75sin (πt / 6) where t is the time in months, with t = 1 corresponding to January. Determine the months when sales exceed 100,000 units.
A sharpshooter intends to hit a target at a distance of 1000 yards (3000 ft) with a gun that has a muzzle velocity of 1200 feet per second (see figure). Neglecting air resistance, determine the minimum angle of elevation of the gun if the range is given by
r = 1/32 v02 sin2θ
The table gives the unemployment rate r for the years 1985 through 1994 in the United States. The time t is measured in years, with t = 0 corresponding to 1990. (Source: U.S. Bureau of Labor Statistics)
t |-5 |-4 |-3 |-2 |-1 |0 |1 |2 |3 |4 | |r |7.2 |7.0 |6.2 |5.5 |5.3 |5.5 |6.7 |7.4 |6.8 |6.1 | |a) Create a scatter plot of the data.
b) Which of the following models best represents the data? Explain your reasoning.
(1) r = 1.5 cos(t + 3.9) + 6.37
(2) r = 1.03 sin(0.9t + 0.44) + 6.19
(3) r = 1.05 sin[0.95(t + 6.32)] + 6.20
(4) r = 1.5 sin[0.5(t + 2.8)]+ 6.25
c) What term in the model gives the average unemployment rate? What is the rate?
d) Economists study the lengths of business cycles such as unemployment rates. Based on this short span of time, use the model to give the length of this cycle.
e) Use the model to estimate the next time the unemployment
rate will be 6% or less.
Approximating
So far we can find EXACT angles for these:
sin θ = [pic] cos θ = [pic] tan θ = 0
Sometimes the best we can do is _________________________ with the calculator with problems like these:
* Use the ___________________________ to find one angle.
* Use ________________________________ to find the other.
sin θ = ¼ cos θ = .7
tan θ = 10 3sin x = 1 - sin x
Verifying Trig Identities
Some "Tips" for Proving an Identity
• Show ____________________ step.
• Start with the most ___________________________ side first.
• Have a "rough draft" and a "final draft."
• _____________ something! You never know where things will lead.
• And you can always try something else if that doesn't work.
• Try working from both ends to connect the "chain."
• Sometimes it helps to put everything in terms of _____ and ______.
Example:
• Look for squares and opportunities to use Pythagorean Identities.
Example:
• Add fractions (need common denominator.)
Example:
• Pull apart fractions.
Example:
• Multiply by a conjugate.
Example:
• Factor.
Example:
Algebra Stuff:
Examples:
Even & Odd Identities
Write as a function of a positive angle.
cos (-19o)=
csc (-19o)=
cot (-19o)=
sec (-19o)=
True or False?
Hint: Remember sin α = sin β if α & β are coterminal.
sin (-30o) = sin (30o)
cos (330o) = cos (30o)
tan (119o) = -tan (241o)
sec (80o) = -sec (280o)
Sum and Difference Formulas
Simplify to a trig function of θ: [pic]
Simplify without using a calculator: [pic]
Find the exact value. Use a sum or difference identity: sin 240°
Solve the trigonometric equation for 0 < θ < 2π. [pic]
Verify [pic]
Verify [pic]
Use the definitions of the trigonometric ratios for a right triangle to derive the cofunction identity: tan (90° - A)
Double Angle & Half Angle Identities
Examples:
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