7.5 Solving trigonometric equations (Page 5/7)

Precalculus

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Solving an equation using an identity

Solve the equation exactly using an identity: $3 \cos θ + 3 = 2 \sin^{2} θ, 0 \leq θ < 2 π .$

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

\begin{matrix} 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 & = - \frac{1}{2} \\ θ & = \frac{2 π}{3}, \frac{4 π}{3} \\ \cos θ + 1 & = 0 \\ \cos θ & = - 1 \\ θ & = π \end{matrix}

Our solutions are $\frac{2 π}{3}, \frac{4 π}{3}, π .$

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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: $\cos (2 x) = \frac{1}{2}$ on $[0, 2 π) .$

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

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

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

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

\begin{array}{l} 2 x = \frac{π}{3} + 2 π \\ = \frac{π}{3} + \frac{6 π}{3} \\ = \frac{7 π}{3} \end{array}

so $x = \frac{7 π}{6} .$

One more rotation yields

\begin{array}{l} \begin{array}{l} 2 x = \frac{π}{3} + 4 π \end{array} \\ = \frac{π}{3} + \frac{12 π}{3} \\ = \frac{13 π}{3} \end{array}

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

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

\begin{array}{l} 2 x = \frac{5 π}{3} + 2 π \\ = \frac{5 π}{3} + \frac{6 π}{3} \\ = \frac{11 π}{3} \end{array}

so $x = \frac{11 π}{6} .$

One more rotation yields

\begin{array}{l} 2 x = \frac{5 π}{3} + 4 π \\ = \frac{5 π}{3} + \frac{12 π}{3} \\ = \frac{17 π}{3} \end{array}

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

Our solutions are $\frac{π}{6}, \frac{5 π}{6}, \frac{7 π}{6}, and \frac{11 π}{6} .$ 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.

\begin{array}{l} \begin{array}{l} \begin{array}{l} a^{2} + b^{2} = c^{2} \end{array} \\ {(23)}^{2} + {(69.5)}^{2} \approx 5359 \\ \sqrt{5359} \approx 73.2 m \end{array} \end{array}

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.

\begin{array}{l} \tan θ = \frac{69.5}{23} \\ \tan^{- 1} (\frac{69.5}{23}) \approx 1.2522 \\ \approx {71.69}^{\circ} \end{array}

The angle of elevation is approximately ${71.7}^{\circ},$ and the length of the cable is 73.2 meters.

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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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