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Trigonometry : Law of Sines

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Example Questions

Example Question #51 : Law Of Cosines And Law Of Sines

If b=25 , find the remaining angles and sides. Los 5

Possible Answers:

$\\ A=62^\circ \\ a=27.4\:\uptext{ft} \\ c=48.3\:\uptext{ft}$

$\\ A=62^\circ \\ a=50\:\uptext{ft} \\ c=48.3\:\uptext{ft}$

$\\ A=45^\circ \\ a=42.3\:\uptext{ft} \\ c=48.3\:\uptext{ft}$

$\\ A=45^{\circ} \\ a=35.4\:\uptext{ft} \\ c=48.3\:\uptext{ft}$

None of the other answers.

Correct answer:

$\\ A=45^{\circ} \\ a=35.4\:\uptext{ft} \\ c=48.3\:\uptext{ft}$

Explanation:

The Law of Sines is a set of ratios that allows one to compute missing angles and side lengths of oblique triangles (non right angle triangles).

The Law of Sines:

$\frac{a}{\sin A}=\frac{b}{\sin B}=\frac{c}{\sin C}$ .

To find the missing angle A we subtract the sum of the two known angles from 180° as the interior angles of all triangles equal 180°.

$180^{\circ}-135^{\circ}=45^{\circ}$

The LOS can be rearranged to solve for the missing side.

$\frac{a}{\sin A}=\frac{b}{\sin B}\rightarrow a=\frac{b}{\sin B}\cdot\sin A\rightarrow \frac{25}{\sin 30}\cdot\sin 45=35.4ft$

$\frac{b}{\sin B}=\frac{c}{\sin C}\rightarrow c=\frac{b}{\sin B}\cdot\sin C\rightarrow c=\frac{25}{\sin 30}\cdot\sin 105=48.3ft$

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Example Question #21 : Law Of Sines

If A = 39 , B = 100 , and b = 19 determine the length of side , round to the nearest whole number.

Possible Answers:

Correct answer:

Explanation:

This is a straightforward Law of Sines problem as we are given two angles and a corresponding side:

$\frac{a}{\sin A} = \frac{b}{\sin B}$

Substituting the known values:

$\frac{a}{\sin 39} = \frac{19}{\sin 100}$

Solving for the unknown side:

$a = \sin 39 \left(\frac{19}{\sin 100} \right ) = 12.14 = 12$

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Example Question #51 : Law Of Cosines And Law Of Sines

If $B = 72^{\circ}$ , a = 100 , and b = 802 determine the measure of $\angle A$ , round to the nearest degree.

Possible Answers:

$6^{\circ}$

$72^{\circ}$

$7^{\circ}$

$8^{\circ}$

$91^{\circ}$

Correct answer:

$7^{\circ}$

Explanation:

This is a straightforward Law of Sines problem since we are given one angle and two sides and are asked to determine the corresponding angle.

$\frac{a}{\sin A} = \frac{b}{\sin B}$

Substituting the given values:

$\frac{100}{\sin A} = \frac{802}{\sin 72^{\circ}}$

Now rearranging the equation:

$\sin A = \frac{100\sin 72^{\circ}}{802}$

The final step is to take the inverse sine of both sides:

$A = \sin^{-1}\left(\frac{100\sin 72^{\circ}}{802}\right) = 6.81^{\circ} = 7^{\circ}$

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Example Question #21 : Law Of Sines

If $\small b=4$ , $\small \angle B$ = $\small 2^o$ , and $\small \small \angle A$ = $\small 40^o$ , find the length of side $\small a$ .

Possible Answers:

$\small 19$

$\small 74$

$\small 14$

$\small 26$

Correct answer:

$\small 74$

Explanation:

We are given two angles and the length of the corresponding side to one of those angles. Because the problem is asking for the corresponding length of the other angle we can use the Law of Sines to find the length of the side $\small a$ . The equation for the Law of Sines is

$\small \frac{a}{\sin\angle A} = \frac{b}{\sin\angle B}$

If we rearrange the equation to isolate we obtain

$\small a = \frac{b\sin\angle A}{\sin\angle B}$

Substituting on the values given in the problem

$\small a = \frac{4\sin40^o}{\sin2^o}$

$\small a = 74$

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Example Question #24 : Law Of Sines

If $\small c=18$ , $\small \small \angle B$ = $\small 22^o$ , and $\small \angle C$ = $\small 7^o$ , find the length of side $\small b$ to the nearest whole number.

Possible Answers:

$\small 30$

$\small 55$

$\small 71$

$\small 63$

Correct answer:

$\small 55$

Explanation:

Because we are given the two angles and the length of the corresponding side to one of those angles, we can use the Law of Sines to find the length of the side that we need. So we use the equation

$\small \frac{b}{\sin\angle B} = \frac{c}{\sin\angle C}$

Rearranging the equation to isolate $\small b$ gives

$\small b = \frac{c\sin\angle B}{\sin\angle C}$

Substituting in the values from the problem gives

$\small b = \frac{18\sin22^o}{\sin7^o}$

$\small b = 55$

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Example Question #21 : Law Of Sines

If $\small \angle A = 40^o$ , $\small b = 3.5$ , and $\small \angle B = 35^o$ , find to the nearest whole number.

Possible Answers:

$\small 4$

$\small 8$

$\small 7$

$\small 11$

Correct answer:

$\small 4$

Explanation:

We can use the Law of Sines to find the length of the missing side, because we have its corresponding angle and the length and angle of another side. The equation for the Law of Sines is

$\small \frac{a}{\sin\angle A} = \frac{b}{\sin\angle B}$

Isolating $\small a$ gives us

$\small a = \frac{b\sin\angle A}{\sin\angle B}$

Finally, substituting in the values of the of from the problem gives

$\small a= \frac{3.5\sin40^o}{\sin35^o}$

$\small a = 4$

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Trigonometry : Law of Sines

Study concepts, example questions & explanations for Trigonometry

All Trigonometry Resources

Example Questions

Example Question #51 : Law Of Cosines And Law Of Sines

Example Question #21 : Law Of Sines

Example Question #51 : Law Of Cosines And Law Of Sines

Example Question #21 : Law Of Sines

Example Question #24 : Law Of Sines

Example Question #21 : Law Of Sines

All Trigonometry Resources

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