GRAPHICAL REASONING Use a graphing utility to confirm the

sin2 x

sin2 x 4 sin x  6 0 s

sin2 x 5 sin x 6 0 si

When solving a trigonometric equation, the preliminary goal is to ________ the trigonometric function involved in the equation.

The equation has the solutions and which are called ________ solutions.

The equation is a trigonometric equation that is of ________ type.

A solution of an equation that does not satisfy the original equation is called an ________ solution.

(a) (b)

(a) (b)

3 tan2 2x 1  0

2 cos2 4x 1  0

2 sin2 x sin x 1  0 x

csc4 x 4 csc2 x  0

2 cos x 1  0

2 sin x 1  0

3 csc x 2  0 ta

tan x 3  0

3 sec

3 cot2

sin x sin x 1  0

3 tan2 x 1 tan2 x 3  0 s

4 cos x 2 x 1  0

sin2 x  3 cos2 4 cos x

2 sin 3x  3 2 2x  1

tan2 2 sin 3x  3 2

tan 3x tan x 1  0 c

cos 2x 2 cos x 1  0

cos

sec2 cos x 1  0 3

3 tan

2 sin2

sec

sec x csc x  2 csc x

2 sin x csc x  0

sec x tan x  1

2 cos2 x cos x 1  0

2 sin2 x 3 sin x 1  0

2 sec2 x tan2 x 3  0

cos x sin x tan x  2

csc x cot x  1

sin x 2  cos x 2 c

cos 2x  1 2

sin 2x  3 2 c

tan 3x  1 s

sec 4x  2

cos x 2  2 2

sin x 2  3 2 co

y  sin

y  sin x cos x x

y  tan

y  sec4  x 8 y  tan  4

sin x cos x cos x 1 sin x  4

cos x cot x 1 sin x  3

x tan x 1  0 x

x cos x 1  0

sec2 x 0.5 tan x 1  0

csc2 x 0.5 cot x 5  0

2 tan2 x 7 tan x 15  0

6 sin2 x 7 sin x 2  0

12 sin2 x 13 sin x 3  0

3 tan2 x 4 tan x 4  0

tan2 x 3 tan x 1  0

4 cos2 x 4 cos x 1  0 t

tan2 x tan x 12  0

tan2 x tan x 2  0 t

tan2 x 6 tan x 5  0

sec2 x tan x 3  0

2 cos2 x 5 cos x 2  0

2 sin2 x 7 sin x 3  0

cot2 x 9  0

cot2 x 6 cot x 5  0

sec2 x 4 sec x  0

sec2 x 2 sec x 8  0

csc2 x 3 csc x 4  0

csc2 x 5 csc x  0

3 tan2 x 5 tan x 4  0,

cos 0, 2 x 2 cos x 1  0,

4 cos2 x 2 sin x 1  0,

2 sec2 x tan x 6  0,

f x  sin 2 sin x cos x sin x  0 2 x cos x

f x  cos 2 sin x cos x cos x  0 2 x sin x f

f x  sin x cos x cos x sin x  0

f x  2 sin x cos 2x 2 cos x 4 sin x cos x  0

f x  sin x cos x x  0

f x  sec x tan x x

f x  tan

f x  cos x

GRAPHICAL REASONING Consider the function given by and its graph shown in the figure. (a) What is the domain of the function? (b) Identify any symmetry and any asymptotes of the graph. (c) Describe the behavior of the function as (d) How many solutions does the equation have in the interval Find the solutions. (e) Does the equation have a greatest solution? If so, approximate the solution. If not, explain why

GRAPHICAL REASONING Consider the function given by and its graph shown in the figure. (a) What is the domain of the function? (b) Identify any symmetry and any asymptotes of the graph. (c) Describe the behavior of the function as (d) How many solutions does the equation have in the interval Find the solutions.

HARMONIC MOTION A weight is oscillating on the end of a spring (see figure). The position of the weight relative to the point of equilibrium is given by where is the displacement (in meters) and is the time (in seconds). Find the times when the weight is at the point of equilibrium

DAMPED HARMONIC MOTION The displacement from equilibrium of a weight oscillating on the end of a spring is given by where is the displacement (in feet) and is the time (in seconds). Use a graphing utility to graph the displacement function for Find the time beyond which the displacement does not exceed 1 foot from equilibrium.

SALES The monthly sales (in thousands of units) of a seasonal product are approximated by where is the time (in months), with corresponding to January. Determine the months in which sales exceed 100,000 units.

SALES The monthly sales (in hundreds of units) of skiing equipment at a sports store are approximated by where is the time (in months), with corresponding to January. Determine the months in which sales exceed 7500 units

PROJECTILE MOTION A batted baseball leaves the bat at an angle of with the horizontal and an initial velocity of feet per second. The ball is caught by an outfielder 300 feet from home plate (see figure). Find if the range of a projectile is given by

PROJECTILE MOTION A sharpshooter intends to hit a target at a distance of 1000 yards with a gun that has a muzzle velocity of 1200 feet per second (see figure). Neglecting air resistance, determine the guns minimum angle of elevation if the range is given by

FERRIS WHEEL A Ferris wheel is built such that the height (in feet) above ground of a seat on the wheel at time (in minutes) can be modeled by The wheel makes one revolution every 32 seconds. The ride begins when (a) During the first 32 seconds of the ride, when will a person on the Ferris wheel be 53 feet above ground? (b) When will a person be at the top of the Ferris wheel for the first time during the ride? If the ride lasts 160 seconds, how many times will a person be at the top of the ride, and at what times?

DATA ANALYSIS: METEOROLOGY The table shows the average daily high temperatures in Houston (in degrees Fahrenheit) for month with corresponding to January.

GEOMETRY The area of a rectangle (see figure) inscribed in one arc of the graph of is given by (a) Use a graphing utility to graph the area function, and approximate the area of the largest inscribed rectangle. (b) Determine the values of for which

QUADRATIC APPROXIMATION Consider the function given by (a) Approximate the zero of the function in the interval

The equation has four times the number of solutions in the interval as the equation

If you correctly solve a trigonometric equation to the statement then you can finish solving the equation by using an inverse function.

THINK ABOUT IT Explain what would happen if you divided each side of the equation by Is this a correct method to use when solving equations?

GRAPHICAL REASONING Use a graphing utility to confirm the solutions found in Example 6 in two different ways. (a) Graph both sides of the equation and find the coordinates of the points at which the graphs intersect. Left side: Right side: (b) Graph the equation and find the intercepts of the graph. Do both methods produce the same values? Which method do you prefer? Explain.

Explain in your own words how knowledge of algebra is important when solving trigonometric equations.

CAPSTONE Consider the equation Explain the similarities and differences between finding all solutions in the interval finding all solutions in the interval and finding the general solution. 0