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Solving a linear equation involving the sine function

Find all possible exact solutions for the equation sin t = 1 2 .

Solving for all possible values of t means that solutions include angles beyond the period of 2 π . From [link] , we can see that the solutions are π 6 and 5 π 6 . But the problem is asking for all possible values that solve the equation. Therefore, the answer is

π 6 ± 2 π k   and   5 π 6 ± 2 π k

where k is an integer.

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Given a trigonometric equation, solve using algebra .

  1. Look for a pattern that suggests an algebraic property, such as the difference of squares or a factoring opportunity.
  2. Substitute the trigonometric expression with a single variable, such as x or u .
  3. Solve the equation the same way an algebraic equation would be solved.
  4. Substitute the trigonometric expression back in for the variable in the resulting expressions.
  5. Solve for the angle.

Solve the trigonometric equation in linear form

Solve the equation exactly: 2 cos θ 3 = 5 , 0 θ < 2 π .

Use algebraic techniques to solve the equation.

2 cos θ 3 = 5         2 cos θ = 2           cos θ = 1                θ = π
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Solve exactly the following linear equation on the interval [ 0 , 2 π ) : 2 sin x + 1 = 0.

x = 7 π 6 , 11 π 6

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Solving equations involving a single trigonometric function

When we are given equations that involve only one of the six trigonometric functions, their solutions involve using algebraic techniques and the unit circle (see [link] ). We need to make several considerations when the equation involves trigonometric functions other than sine and cosine. Problems involving the reciprocals of the primary trigonometric functions need to be viewed from an algebraic perspective. In other words, we will write the reciprocal function, and solve for the angles using the function. Also, an equation involving the tangent function is slightly different from one containing a sine or cosine function. First, as we know, the period of tangent is π , not 2 π . Further, the domain of tangent is all real numbers with the exception of odd integer multiples of π 2 , unless, of course, a problem places its own restrictions on the domain.

Solving a problem involving a single trigonometric function

Solve the problem exactly: 2 sin 2 θ 1 = 0 , 0 θ < 2 π .

As this problem is not easily factored, we will solve using the square root property. First, we use algebra to isolate sin θ . Then we will find the angles.

2 sin 2 θ 1 = 0        2 sin 2 θ = 1          sin 2 θ = 1 2        sin 2 θ = ± 1 2            sin θ = ± 1 2 = ± 2 2                 θ = π 4 , 3 π 4 , 5 π 4 , 7 π 4
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Solving a trigonometric equation involving cosecant

Solve the following equation exactly: csc θ = 2 , 0 θ < 4 π .

We want all values of θ for which csc θ = 2 over the interval 0 θ < 4 π .

csc θ = 2 1 sin θ = 2 sin θ = 1 2       θ = 7 π 6 , 11 π 6 , 19 π 6 , 23 π 6
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Solving an equation involving tangent

Solve the equation exactly: tan ( θ π 2 ) = 1 , 0 θ < 2 π .

Recall that the tangent function has a period of π . On the interval [ 0 , π ) , and at the angle of π 4 , the tangent has a value of 1. However, the angle we want is ( θ π 2 ) . Thus, if tan ( π 4 ) = 1 , then

θ π 2 = π 4 θ = 3 π 4 ± k π

Over the interval [ 0 , 2 π ) , we have two solutions:

3 π 4   and  3 π 4 + π = 7 π 4
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Questions & Answers

A golfer on a fairway is 70 m away from the green, which sits below the level of the fairway by 20 m. If the golfer hits the ball at an angle of 40° with an initial speed of 20 m/s, how close to the green does she come?
Aislinn Reply
cm
tijani
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John Reply
what is physics
Siyaka Reply
A mouse of mass 200 g falls 100 m down a vertical mine shaft and lands at the bottom with a speed of 8.0 m/s. During its fall, how much work is done on the mouse by air resistance
Jude Reply
Can you compute that for me. Ty
Jude
what is the dimension formula of energy?
David Reply
what is viscosity?
David
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emma Reply
what is chemistry
Youesf Reply
what is inorganic
emma
Chemistry is a branch of science that deals with the study of matter,it composition,it structure and the changes it undergoes
Adjei
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Adjanou
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Pedro
A ball is thrown straight up.it passes a 2.0m high window 7.50 m off the ground on it path up and takes 1.30 s to go past the window.what was the ball initial velocity
Krampah Reply
2. A sled plus passenger with total mass 50 kg is pulled 20 m across the snow (0.20) at constant velocity by a force directed 25° above the horizontal. Calculate (a) the work of the applied force, (b) the work of friction, and (c) the total work.
Sahid Reply
you have been hired as an espert witness in a court case involving an automobile accident. the accident involved car A of mass 1500kg which crashed into stationary car B of mass 1100kg. the driver of car A applied his brakes 15 m before he skidded and crashed into car B. after the collision, car A s
Samuel Reply
can someone explain to me, an ignorant high school student, why the trend of the graph doesn't follow the fact that the higher frequency a sound wave is, the more power it is, hence, making me think the phons output would follow this general trend?
Joseph Reply
Nevermind i just realied that the graph is the phons output for a person with normal hearing and not just the phons output of the sound waves power, I should read the entire thing next time
Joseph
Follow up question, does anyone know where I can find a graph that accuretly depicts the actual relative "power" output of sound over its frequency instead of just humans hearing
Joseph
"Generation of electrical energy from sound energy | IEEE Conference Publication | IEEE Xplore" ***ieeexplore.ieee.org/document/7150687?reload=true
Ryan
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Maurice Reply
what are the types of wave
Maurice
answer
Magreth
progressive wave
Magreth
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hi
Mujahid
A string is 3.00 m long with a mass of 5.00 g. The string is held taut with a tension of 500.00 N applied to the string. A pulse is sent down the string. How long does it take the pulse to travel the 3.00 m of the string?
yasuo Reply
Who can show me the full solution in this problem?
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Source:  OpenStax, Precalculus. OpenStax CNX. Jan 19, 2016 Download for free at https://legacy.cnx.org/content/col11667/1.6
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