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In American Football, the quarterback tries to throw the ball to a wide receiver. The equation to model this is \(R = \frac{v^2 \sin 2θ}{32}\) where R is the distance in feet, v is the speed the ball is thrown in feet/second, and θ is the angle above the horizontal that the ball is thrown. If the distance to the receiver and the speed of the throw are known, then the angle that the quarterback has to throw the ball can be calculated. Solve Trigonometric EquationsThe general method to solve trigonometric equations is to isolate a trigonometric expression. This can be done by factoring, using trigonometric identities, or other problem solving techniques. Several methods to isolate the trigonometric expression are:
After the trigonometric expression is isolated, find the angles that satisfy the trigonometric expression. Remember that each trigonometric function is positive in two quadrants and negative in two quadrants, so there are usually at least two solutions for each trigonometric equation. Because trigonometric functions are periodic, there are an infinite number of solutions that can be found by adding the period of the function. Find the Solutions of Trigonometric Equations After the trigonometric expression is isolated, find the angles that satisfy the trigonometric expression. If the results are on the unit circle,
Example 1: Solve a Simple Trigonometric EquationFind all solutions of \(2 \sin x - 1 = 0\). SolutionThere is only one trigonometric function, so isolate it and solve. $$2 \sin x - 1 = 0$$ $$2 \sin x = 1$$ $$\sin x = \frac{1}{2}$$ Think of the angles on the unit circle where \(\sin x = \frac{1}{2}\). Figure 2: Angles where \(\sin x = \frac{1}{2}\)The angles are \(x = \frac{π}{6}, \frac{5π}{6}\). These are not spaced evenly, so add the period times n. All the solutions are \(x = \frac{π}{6} + 2πn\) and \(\frac{5π}{6} + 2πn\). Try It 1Solve \(2 \cos α + \sqrt{2} = 0\). Answer\(\frac{3π}{4} + 2πn, \frac{5π}{4} + 2πn\) Example 2: Solve a Trigonometric Equation by FactoringFind all solutions of \(\sin x \tan x + \sin x = 0\). SolutionBegin by factoring out the common factor, sin x. $$\sin x \tan x + \sin x = 0$$ $$\sin x\left(\tan x + 1\right) = 0$$ Apply the zero product property and set each factor equal to zero and solve.
The solutions are \(x = πn, \frac{3π}{4} + πn\). Try It 2Solve \(\tan x \sec x - \tan x = 0\). Answerπn Example 3: Solve a Trigonometric Equation by FactoringFind all solutions of \(2 \cos^2 α + \cos α - 1 = 0\). SolutionThis is a quadratic type equation, so factor it as you would quadratic trinomial. $$2 \cos^2 α + \cos α - 1 = 0$$ $$\left(2 \cos α - 1\right)\left(\cos α + 1\right) = 0$$ Set each factor equal to zero and solve. These solutions are evenly spaced around the unit circle with \(\frac{2π}{3}\) between each angle, so the solutions can be written as \(α = \frac{π}{3} + \frac{2πn}{3}\). Try It 3Solve 2 sin2 x − sin x − 1 = 0. Answer\(\frac{π}{2} + \frac{2πn}{3}\) Example 4: Solve a Trigonometric Equation Using an IdentityFind all the solutions on the interval [0, 2π) of \(\sec^2 β + \tan^2 β = 7\). SolutionA Pythagorean identity allows the substitution of \(\tan^2 β + 1\) for \(\sec^2 β\). $$\sec^2 β + \tan^2 β = 7$$ $$\tan^2 β + 1 + \tan^2 β = 7$$ $$2\tan^2 β + 1 = 7$$ $$2\tan^2 β = 6$$ $$\tan^2 β = 3$$ $$\tan β = ±\sqrt{3}$$ $$β = \frac{π}{3}, \frac{2π}{3}, \frac{4π}{3}, \frac{5π}{3}$$ $$β = \frac{π}{3} + πn, \frac{2π}{3} + πn$$ Since only solutions between 0 and 2π are asked for, the solutions can be found by plugging in integers for n starting with 0 until the results are 2π or larger. This gives solutions of \(β = \frac{π}{3}, \frac{2π}{3}, \frac{4π}{3}, \frac{5π}{3}\). Try It 4Solve on the interval [0, 2π): sin2 x − cos2 x = 0. Answer\(\frac{π}{4}, \frac{3π}{4}, \frac{5π}{4}, \frac{7π}{4}\) Example 5: Solve a Trigonometric Equation Using an IdentityFind all the solutions on the interval [0, 2π) of \(\sec θ \cot θ = 1\). SolutionBecause the trigonometric expressions do not equal 0, we cannot use the zero product property, so the expression will have to be simplified. Try rewriting the expression in terms of sine and cosine. $$\sec 2θ \cot 2θ = 1$$ $$\frac{1}{\cos 2θ} \frac{\cos 2θ}{\sin 2θ} = 1$$ $$\frac{1}{\sin 2θ} = 1$$ $$1 = \sin 2θ$$ The angle in the equation is 2θ. $$2θ = \frac{π}{2} + 2πn$$ $$θ = \frac{π}{4} + πn$$ Plug in integers starting with 0 for n until the results are 2π or higher. $$θ = \frac{π}{4}, \frac{5π}{4}$$ Try It 5Solve on the interval [0, 2π): \(2\cos 3x = \sqrt{3}\). Answer\(\frac{π}{18}\) \(\frac{11π}{18}\), \(\frac{13π}{18}\), \(\frac{23π}{18}\), \(\frac{25π}{18}\), \(\frac{35π}{18}\) Example 6: Solve a Trigonometric Equation by Squaring Both SidesFind all the solutions on the interval [0, 2π) of \(\csc 3x + \cot 3x = 1\). SolutionCosecant, cotangent, and 1 are related with a Pythagorean identity, but the problem does not have squares on the trigonometric functions. To fix this, move the cotangent to other side of the equation and square both sides. $$\csc 3x + \cot 3x = 1$$ $$\csc 3x = 1 - \cot 3x$$ $$\csc^2 3x = \left(1 - \cot 3x\right)^2$$ $$\csc^2 3x = 1 - 2\cot 3x + \cot^2 3x$$ Apply the Pythagorean identity for cosecant and cotangent to the csc2 3x. That way cotangent will be the only trigonometric function in the equation. $$1 + \cot^2 3x = 1 - 2\cot 3x + \cot^2 3x$$ Subtract 1 and cot2 3x from both sides. $$0 = -2\cot 3x$$ $$0 = \cot 3x$$ The angle in the equation is 3x. $$3x = \frac{π}{2} + πn$$ $$x = \frac{π}{6} + \frac{πn}{3}$$ Plug in integers starting with 0 for n until the results are 2π or higher. $$x = \frac{π}{6}, \frac{π}{2}, \frac{5π}{6}, \frac{7π}{6}, \frac{3π}{2}, \frac{11π}{6}$$ Try It 6Solve on the interval [0, 2π): cos 2x – sin 2x = 1. Answer0, \(\frac{3π}{4}\), π, \(\frac{7π}{4}\) Example 7: Solve a Trigonometric Equation by GraphingUse a graphing utility to find all the solutions in the interval [0, 2π) to three decimal places of tan x = csc x. SolutionMake the equation equal 0, tan x − csc x = 0. Then graph the expression. Use the calculate zero feature to find the x-intercepts. On a TI graphing calculator, press Y= and type in the expression. To enter csc x, type \(\frac{1}{\sin x}\). Then press ZOOM and choose ZTrig from the menu. This gives a nice graph on the interval [−2π, 2π]. To find the x-intercepts, press 2nd CALC and choose zero from the menu. Use the left and right arrow keys to move the cursor to the left of an intercept and press enter. Then move the cursor to the right of the intercept and press enter. Finally move the cursor to the approximate intercept to make a guess and press enter. The intercept will be displayed on the bottom of the screen. On a NumWorks calculator, choose Grapher from the Home screen. Enter the equation after it is solved for 0, tan x − csc x = 0. Enter csc x as \(\frac{1}{\sin x}\). Go to the Graph tab. Press OK. Select Find, then Zeros. Use the left and right arrow keys to see all the zeros. Figure 5: y = tan x − csc xThe x-intercepts and solutions are x = 0.905 and x = 5.379. Try It 7Solve on the interval [0, 2π) using a graphing utility: sin2 x + sin x = 0. Answer0, 3.142, 4.712 Lesson SummarySeveral methods to isolate the trigonometric expression are:
Find the Solutions of Trigonometric Equations After the trigonometric expression is isolated, find the angles that satisfy the trigonometric expression. If the results are on the unit circle,
Practice Exercises
Find all solutions of the following equations. Find all solutions in the interval [0, 2π). Use a graphing utility to find the solutions in the interval [0, 2π) to three decimal places. Problem Solving Mixed Review Answers
What are the solutions in the interval 0 ≤ θ 2π?The solutions within the domain 0 ≤ θ < 2 π 0 ≤ θ < 2 π are 0 , π , 7 π 6 , 11 π 6 .
Does 0 2pi include 0?All numbers between 0 and 2π , including the 0 and 2π , are included.
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