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Solve trigonometric equations using a calculator

Not all functions can be solved exactly using only the unit circle. When we must solve an equation involving an angle other than one of the special angles, we will need to use a calculator. Make sure it is set to the proper mode, either degrees or radians, depending on the criteria of the given problem.

Using a calculator to solve a trigonometric equation involving sine

Use a calculator to solve the equation sin θ = 0.8 , where θ is in radians.

Make sure mode is set to radians. To find θ , use the inverse sine function. On most calculators, you will need to push the 2 ND button and then the SIN button to bring up the sin 1 function. What is shown on the screen is sin 1 ( . The calculator is ready for the input within the parentheses. For this problem, we enter sin 1 ( 0.8 ) , and press ENTER. Thus, to four decimals places,

sin 1 ( 0.8 ) 0.9273

The solution is

θ 0.9273 ± 2 π k

The angle measurement in degrees is

θ 53.1° θ 180° 53.1° 126.9°
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Using a calculator to solve a trigonometric equation involving secant

Use a calculator to solve the equation sec θ = −4 , giving your answer in radians.

We can begin with some algebra.

sec θ = 4 1 cos θ = 4 cos θ = 1 4

Check that the MODE is in radians. Now use the inverse cosine function.

cos 1 ( 1 4 ) 1.8235 θ 1.8235 + 2 π k

Since π 2 1.57 and π 3.14 , 1.8235 is between these two numbers, thus θ 1 .8235 is in quadrant II. Cosine is also negative in quadrant III. Note that a calculator will only return an angle in quadrants I or II for the cosine function, since that is the range of the inverse cosine. See [link] .

Graph of angles theta =approx 1.8235, theta prime =approx pi - 1.8235 = approx 1.3181, and then theta prime = pi + 1.3181 = approx 4.4597

So, we also need to find the measure of the angle in quadrant III. In quadrant III, the reference angle is θ ' π 1 .8235 1 .3181 . The other solution in quadrant III is θ ' π + 1 .3181 4 .4597 .

The solutions are θ 1.8235 ± 2 π k and θ 4.4597 ± 2 π k .

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Solve cos θ = 0.2.

θ 1.7722 ± 2 π k and θ 4.5110 ± 2 π k

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Solving trigonometric equations in quadratic form

Solving a quadratic equation    may be more complicated, but once again, we can use algebra as we would for any quadratic equation. Look at the pattern of the equation. Is there more than one trigonometric function in the equation, or is there only one? Which trigonometric function is squared? If there is only one function represented and one of the terms is squared, think about the standard form of a quadratic. Replace the trigonometric function with a variable such as x or u . If substitution makes the equation look like a quadratic equation, then we can use the same methods for solving quadratics to solve the trigonometric equations.

Solving a trigonometric equation in quadratic form

Solve the equation exactly: cos 2 θ + 3 cos θ 1 = 0 , 0 θ < 2 π .

We begin by using substitution and replacing cos θ with x . It is not necessary to use substitution, but it may make the problem easier to solve visually. Let cos θ = x . We have

x 2 + 3 x 1 = 0

The equation cannot be factored, so we will use the quadratic formula     x = b ± b 2 4 a c 2 a .

x = 3 ± ( 3 ) 2 4 ( 1 ) ( 1 ) 2 = 3 ± 13 2

Replace x with cos θ , and solve.

cos θ = 3 ± 13 2 θ = cos 1 ( 3 + 13 2 )

Note that only the + sign is used. This is because we get an error when we solve θ = cos 1 ( 3 13 2 ) on a calculator, since the domain of the inverse cosine function is [ 1 , 1 ] . However, there is a second solution:

θ = cos 1 ( 3 + 13 2 ) 1.26

This terminal side of the angle lies in quadrant I. Since cosine is also positive in quadrant IV, the second solution is

θ = 2 π cos 1 ( 3 + 13 2 )   5.02
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Questions & Answers

f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
Ken Reply
proof
AUSTINE
sebd me some questions about anything ill solve for yall
Manifoldee Reply
how to solve x²=2x+8 factorization?
Kristof Reply
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
SO THE ANSWER IS X=-8
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
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Find the possible value of 8.5 using moivre's theorem
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which of these functions is not uniformly cintinuous on (0, 1)? sinx
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which of these functions is not uniformly continuous on 0,1
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solve this equation by completing the square 3x-4x-7=0
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X=7
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=7
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x=7
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3x-4x-7=0 -x=7 x=-7
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deals with circles, angles, and triangles. Usually in the form of Soh cah toa or sine, cosine, and tangent
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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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