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Solving an equation using an identity

Solve the equation exactly using an identity: 3 cos θ + 3 = 2 sin 2 θ , 0 θ < 2 π .

If we rewrite the right side, we can write the equation in terms of cosine:

3 cos θ + 3 = 2 sin 2 θ 3 cos θ + 3 = 2 ( 1 cos 2 θ ) 3 cos θ + 3 = 2 2 cos 2 θ 2 cos 2 θ + 3 cos θ + 1 = 0 ( 2 cos θ + 1 ) ( cos θ + 1 ) = 0 2 cos θ + 1 = 0 cos θ = 1 2 θ = 2 π 3 , 4 π 3 cos θ + 1 = 0 cos θ = 1 θ = π

Our solutions are θ = 2 π 3 , 4 π 3 , π .

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Solving trigonometric equations with multiple angles

Sometimes it is not possible to solve a trigonometric equation with identities that have a multiple angle, such as sin ( 2 x ) or cos ( 3 x ) . When confronted with these equations, recall that y = sin ( 2 x ) is a horizontal compression    by a factor of 2 of the function y = sin x . On an interval of 2 π , we can graph two periods of y = sin ( 2 x ) , as opposed to one cycle of y = sin x . This compression of the graph leads us to believe there may be twice as many x -intercepts or solutions to sin ( 2 x ) = 0 compared to sin x = 0. This information will help us solve the equation.

Solving a multiple angle trigonometric equation

Solve exactly: cos ( 2 x ) = 1 2 on [ 0 , 2 π ) .

We can see that this equation is the standard equation with a multiple of an angle. If cos ( α ) = 1 2 , we know α is in quadrants I and IV. While θ = cos 1 1 2 will only yield solutions in quadrants I and II, we recognize that the solutions to the equation cos θ = 1 2 will be in quadrants I and IV.

Therefore, the possible angles are θ = π 3 and θ = 5 π 3 . So, 2 x = π 3 or 2 x = 5 π 3 , which means that x = π 6 or x = 5 π 6 . Does this make sense? Yes, because cos ( 2 ( π 6 ) ) = cos ( π 3 ) = 1 2 .

Are there any other possible answers? Let us return to our first step.

In quadrant I, 2 x = π 3 , so x = π 6 as noted. Let us revolve around the circle again:

2 x = π 3 + 2 π = π 3 + 6 π 3 = 7 π 3

so x = 7 π 6 .

One more rotation yields

2 x = π 3 + 4 π = π 3 + 12 π 3 = 13 π 3

x = 13 π 6 > 2 π , so this value for x is larger than 2 π , so it is not a solution on [ 0 , 2 π ) .

In quadrant IV, 2 x = 5 π 3 , so x = 5 π 6 as noted. Let us revolve around the circle again:

2 x = 5 π 3 + 2 π = 5 π 3 + 6 π 3 = 11 π 3

so x = 11 π 6 .

One more rotation yields

2 x = 5 π 3 + 4 π = 5 π 3 + 12 π 3 = 17 π 3

x = 17 π 6 > 2 π , so this value for x is larger than 2 π , so it is not a solution on [ 0 , 2 π ) .

Our solutions are x = π 6 , 5 π 6 , 7 π 6 , and  11 π 6 . Note that whenever we solve a problem in the form of sin ( n x ) = c , we must go around the unit circle n times.

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Solving right triangle problems

We can now use all of the methods we have learned to solve problems that involve applying the properties of right triangles and the Pythagorean Theorem    . We begin with the familiar Pythagorean Theorem, a 2 + b 2 = c 2 , and model an equation to fit a situation.

Using the pythagorean theorem to model an equation

Use the Pythagorean Theorem, and the properties of right triangles to model an equation that fits the problem.

One of the cables that anchors the center of the London Eye Ferris wheel to the ground must be replaced. The center of the Ferris wheel is 69.5 meters above the ground, and the second anchor on the ground is 23 meters from the base of the Ferris wheel. Approximately how long is the cable, and what is the angle of elevation (from ground up to the center of the Ferris wheel)? See [link] .

Basic diagram of a ferris wheel (circle) and its support cables (form a right triangle). One cable runs from the center of the circle to the ground (outside the circle), is perpendicular to the ground, and has length 69.5. Another cable of unknown length (the hypotenuse) runs from the center of the circle to the ground 23 feet away from the other cable at an angle of theta degrees with the ground. So, in closing, there is a right triangle with base 23, height 69.5, hypotenuse unknown, and angle between base and hypotenuse of theta degrees.

Using the information given, we can draw a right triangle. We can find the length of the cable with the Pythagorean Theorem.

a 2 + b 2 = c 2 ( 23 ) 2 + ( 69.5 ) 2 5359 5359 73.2  m

The angle of elevation is θ , formed by the second anchor on the ground and the cable reaching to the center of the wheel. We can use the tangent function to find its measure. Round to two decimal places.

tan θ = 69.5 23 tan −1 ( 69.5 23 ) 1.2522 71.69°

The angle of elevation is approximately 71.7° , and the length of the cable is 73.2 meters.

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Questions & Answers

sinx sin2x is linearly dependent
cr Reply
what is a reciprocal
Ajibola Reply
The reciprocal of a number is 1 divided by a number. eg the reciprocal of 10 is 1/10 which is 0.1
Shemmy
 Reciprocal is a pair of numbers that, when multiplied together, equal to 1. Example; the reciprocal of 3 is ⅓, because 3 multiplied by ⅓ is equal to 1
Jeza
each term in a sequence below is five times the previous term what is the eighth term in the sequence
Funmilola Reply
I don't understand how radicals works pls
Kenny Reply
How look for the general solution of a trig function
collins Reply
stock therom F=(x2+y2) i-2xy J jaha x=a y=o y=b
Saurabh Reply
sinx sin2x is linearly dependent
cr
root under 3-root under 2 by 5 y square
Himanshu Reply
The sum of the first n terms of a certain series is 2^n-1, Show that , this series is Geometric and Find the formula of the n^th
amani Reply
cosA\1+sinA=secA-tanA
Aasik Reply
Wrong question
Saad
why two x + seven is equal to nineteen.
Kingsley Reply
The numbers cannot be combined with the x
Othman
2x + 7 =19
humberto
2x +7=19. 2x=19 - 7 2x=12 x=6
Yvonne
because x is 6
SAIDI
what is the best practice that will address the issue on this topic? anyone who can help me. i'm working on my action research.
Melanie Reply
simplify each radical by removing as many factors as possible (a) √75
Jason Reply
how is infinity bidder from undefined?
Karl Reply
what is the value of x in 4x-2+3
Vishal Reply
give the complete question
Shanky
4x=3-2 4x=1 x=1+4 x=5 5x
Olaiya
hi can you give another equation I'd like to solve it
Daniel
what is the value of x in 4x-2+3
Olaiya
if 4x-2+3 = 0 then 4x = 2-3 4x = -1 x = -(1÷4) is the answer.
Jacob
4x-2+3 4x=-3+2 4×=-1 4×/4=-1/4
LUTHO
then x=-1/4
LUTHO
4x-2+3 4x=-3+2 4x=-1 4x÷4=-1÷4 x=-1÷4
LUTHO
A research student is working with a culture of bacteria that doubles in size every twenty minutes. The initial population count was  1350  bacteria. Rounding to five significant digits, write an exponential equation representing this situation. To the nearest whole number, what is the population size after  3  hours?
David Reply
f(x)= 1350. 2^(t/20); where t is in hours.
Merkeb

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Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
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