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Trigonometry : Triangles

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Example Questions

Example Question #101 : Triangles

If $\small c=10.3$ , $\small a=7.4$ , and $\small \angle A$ = $\small 34^o$ find $\small \angle C$ to the nearest degree.

Possible Answers:

$\angle C=51^\circ$ and $\angle C'=129^\circ$

$\angle C = 78^\circ$

$\angle C = 51^\circ$ and $\angle C'=78^\circ$

$\angle C = 129^\circ$

$\angle C = 51^\circ$

Correct answer:

$\angle C=51^\circ$ and $\angle C'=129^\circ$

Explanation:

Notice that the given information is Angle-Side-Side, which is the ambiguous case. Therefore, we should test to see if there are no triangles that satisfy, one triangle that satisfies, or two triangles that satisfy this. From $\frac{\sin A}{a}=\frac{\sin C}{c}$ , we get $c\cdot \frac{\sin A}{a}=\sin C$ . In this equation, if $\sin C >1$ , no $\angle A$ that satisfies the triangle can be found. If $\sin C=1, C=90^\circ$ and there is a right triangle determined. Finally, if $\sin C <1$ , two measures of $\angle C$ can be calculated: an acute $\angle C$ and an obtuse angle $C'=180^\circ-C$ . In this case, there may be one or two triangles determined. If $C'+A \geq 180^\circ$ , then the $\angle C'$ is not a solution.

In this problem, $\sin C =.559<1$ , so there may be one or two angles that satisfy this triangle. Since we have the length of two sides of the triangle and the corresponding angle of one of the sides, we can use the Law of Sines to find the angle that we are looking for. This goes as follows:

$\small \small \frac{c}{\sin\angle C} = \frac{a}{\sin\angle A}$

Inputting the values of the problem

$\small \frac{10.3}{\sin\angle C} = \frac{7.4}{\sin\angle 34^o}$

Rearranging the equation to isolate $\small \angle C$

$\small \angle C = \sin^{-1}(\frac{10.3\sin\angle34^o}{7.4})$

$\small \angle C =$ $\small 51^o$

When the original given angle ( $\angle A=34^\circ$ ) is acute, there will be:

One solution if the side opposite the given angle is equal to or greater than the other given side
No solution, one solution (right triangle), or two solutions if the side opposite the given angle is less than the other given side

In this problem, the side opposite the given angle is a=7.4 , which is less than the other given side c=10.3 . Therefore, we have a second solution. Find it by following the below steps:

$\angle C'=180^\circ-C=180^\circ-51^\circ=129^\circ$ .

$C'+A=129^\circ+34^\circ=163^\circ\leq 180^\circ$ , so $\angle C'$ is a solution.

Therefore there are two values for an angle, $\angle C=51^\circ$ and $\angle C'=129^\circ$

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Example Question #6 : Ambiguous Triangles

If c=10.3, a=7.4, and $\angle A=122^\circ$ find $\angle C$ to the nearest degree.

Possible Answers:

$\angle C = 51^\circ$

$\angle C=51^\circ$ and $\angle C'=129^\circ$

No solution

$\angle C = 129^\circ$

Correct answer:

No solution

Explanation:

In this problem, $\sin C = 1.18>1$ , which means that there are no solutions to $\angle C$ that satisfy this triangle. If you got answers for this triangle, check that you set up your Law of Sines equation properly at the start of the problem.

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Example Question #7 : Ambiguous Triangles

If $\small c=10.3$ , a=13 , and $\small \angle A$ = $\small 34^o$ find $\small \angle C$ to the nearest degree.

Possible Answers:

$\angle C = 26.3^\circ$ and $\angle C' = 206.3^\circ$

No solution

$\angle C = 26.3^\circ$

$\angle C = 206.3^\circ$

Correct answer:

$\angle C = 26.3^\circ$

Explanation:

In this problem, $\sin C =.44<1$ , so there may be one or two angles that satisfy this triangle. Since we have the length of two sides of the triangle and the corresponding angle of one of the sides, we can use the Law of Sines to find the angle that we are looking for. This goes as follows:

$\small \small \frac{c}{\sin\angle C} = \frac{a}{\sin\angle A}$

Inputting the values of the problem

$\frac{10.3}{\sin C}=\frac{13}{\sin 34^\circ}$

Rearranging the equation to isolate $\small \angle C$

$\angle C=\sin^-^1\left (\frac{10.3\sin34^\circ}{13} \right )$

$\angle C = 26.3^\circ$

When the original given angle ( $\angle A=34^\circ$ ) is acute, there will be:

One solution if the side opposite the given angle is equal to or greater than the other given side
No solution, one solution (right triangle), or two solutions if the side opposite the given angle is less than the other given side

In this problem, the side opposite the given angle is a=13 , which is greater than the other given side c=10.3 . Therefore, we have only one solution, $\angle C = 26.3^\circ$ .

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Example Question #8 : Ambiguous Triangles

If $\small c=10.3$ , a=12 , and $\angle A =108^\circ$ find $\small \angle C$ to the nearest degree.

Possible Answers:

$35.28^\circ$

No solution

$35.28^\circ$ and $144.72^\circ$

$54.72^\circ$ and $125.28^\circ$

$54.72^\circ$

Correct answer:

$54.72^\circ$

Explanation:

In this problem, $\sin C =.82<1$ , so there may be one or two angles that satisfy this triangle. Since we have the length of two sides of the triangle and the corresponding angle of one of the sides, we can use the Law of Sines to find the angle that we are looking for. This goes as follows:

$\small \small \frac{c}{\sin\angle C} = \frac{a}{\sin\angle A}$

Inputting the values of the problem

$\frac{12}{\sin 108^\circ}=\frac{10.3}{\sin C}$

Rearranging the equation to isolate $\small \angle C$

$\angle C=\sin^-^1\left (\frac{10.3\sin108^\circ}{12} \right )$

$\angle C=54.72^\circ$

When the original given angle ( $\angle A=108^\circ$ ) is obtuse, there will be:

No solution when the side opposite the given angle is less than or equal to the other given side
One solution if the side opposite the given angle is greater than the other given side

In this problem, the side opposite the given angle is a=12 , which is greater than the other given side c=10.3 . Therefore this problem has one and only one solution, $\angle C=54.72^\circ$

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Example Question #9 : Ambiguous Triangles

If c=18 , a=5 , and $\small \angle A$ = $\small 34^o$ find $\small \angle C$ to the nearest degree.

Possible Answers:

$27.19^\circ$ and $152.81^\circ$

$27.19^\circ$

No solution

$8.94^\circ$ and $171.06^\circ$

$8.94^\circ$

Correct answer:

No solution

Explanation:

In this problem, $\sin C=2.01>1$ , which means that there are no solutions to $\angle C$ that satisfy this triangle. If you got answers for this triangle, check that you set up your Law of Sines equation properly at the start of the problem.

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Example Question #61 : Law Of Cosines And Law Of Sines

If c=70, a=50, and $\angle A=122^\circ$ find $\angle C$ to the nearest degree.

Possible Answers:

$49.89^\circ$ and $130.11^\circ$

$8.11^\circ$

$8.11^\circ$ and $171.89^\circ$

$49.89^\circ$

no solution

Correct answer:

no solution

Explanation:

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Example Question #1 : Finding Angles

A 15 foot plank has one end on the ground and one end 6 feet off the ground. What is the measure of the angle formed by the plank and the ground?

Possible Answers:

$\approx 70^{\circ}$

$\approx 50^{\circ}$

$\approx 24^{\circ}$

$\approx 10^{\circ}$

Correct answer:

$\approx 24^{\circ}$

Explanation:

The length of the plank becomes the hypotenuse of the triangle, while the distance between the plank and the ground becomes the length of one side. To solve for the angle between the plank and the ground, you must find the value of $sin (\angle )$ . The sine of the angle is the value of the opposite side over the hypotenuse, which are values that we know.

$sin(\angle )= \frac{opposite}{hypotenuse} = \frac{6}{15}$

$\angle =sin^{-1} \frac{6}{15} = 24^{\circ}$

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Example Question #102 : Triangles

Two angles in a triangle are $30^\circ$ and $70^\circ$ . What is the measure of the 3rd angle?

Possible Answers:

$100^\circ$

$70^\circ$

There is not enough information to determine the angle measure.

$80^\circ$

$90^\circ$

Correct answer:

$80^\circ$

Explanation:

The sum of the angles of a triangle is 180˚.

Thus, since the sum of our two angles is 100˚, our missing angle must be,

180 - 100 = 80 .

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Example Question #101 : Triangles

If the hypotenuse of a right triangle has a length of 6, and the length of a leg is 2, what is the angle between the hypotenuse and the leg?

Possible Answers:

$4.87^\circ$

$13.54^\circ$

$19.47^\circ$

$70.53^\circ$

$32.12^\circ$

Correct answer:

$70.53^\circ$

Explanation:

The leg must be an adjacent side to the hypotenuse.

Therefore, we can use inverse cosine to solve for the angle.

First write the equation for sine of an angle.

$cos(\theta)=\frac{\textup{Adjacent}}{\textup{Hypotenuse}}$

$\theta=cos^{-1}\left(\frac{\textup{Adjacent}}{\textup{Hypotenuse}}\right)$

Substitute the lengths given and solve for the angle.

$\theta=cos^{-1}\left(\frac{\textup{2}}{\textup{6}}\right)=70.53^\circ$

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Example Question #102 : Triangles

A skateboard ramp made so that the rider can gain sufficient speed before a jump is 15 feet high and the ramp is 17 feet long. What is the measure of the angle $\theta$ between the ramp and the ground?

Possible Answers:

$\theta=20^{\circ}$

None of the other answers.

$\theta=34.2^{\circ}$

$\theta=70.4^{\circ}$

$\theta=62^{\circ}$

Correct answer:

$\theta=62^{\circ}$

Explanation:

For the angle in question we have the opposite side and the hypotenuse given to us. We can use the sine function.

$\sin\theta= \frac{opp}{hyp}\rightarrow\sin\theta=(\frac{15}{17})$

Use the inverse sin to find the measure of an angle between these sides:

$\theta =\sin^{-1}(\frac{15}{17})=62^{\circ}$

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Trigonometry : Triangles

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #101 : Triangles

Example Question #6 : Ambiguous Triangles

Example Question #7 : Ambiguous Triangles

Example Question #8 : Ambiguous Triangles

Example Question #9 : Ambiguous Triangles

Example Question #61 : Law Of Cosines And Law Of Sines

Example Question #1 : Finding Angles

Example Question #102 : Triangles

Example Question #101 : Triangles

Example Question #102 : Triangles

All Trigonometry Resources

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