Use the laws of cosines and sines

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Pre-Calculus › Use the laws of cosines and sines

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Which of the following are the missing sides of the triangle?

$\text{I. }4.55$

$\text{II. }5.49$

$\text{III. }1.72$

$\text{IV. }14.56$

$\text{I and III}$

CORRECT

$\text{II and IV}$

$\text{I and II}$

None of the above

Explanation

In order to solve this problem, we need to find

Since all the angles of a triangle add to $180^{\circ}$ , we can easily find it:

$180^{\circ}-65^{\circ}-20^{\circ}=95^{\circ}$

We can now use the Law of Sines to find the missing sides:

$\frac{a}{\sin 65}=\frac{5}{\sin 95}$

$a=\frac{5\times \sin 65}{\sin 95}=4.55$ which is I.

$\frac{b}{\sin 20}=\frac{5}{\sin 95}$

$b=\frac{5\times \sin 20}{\sin 95}=1.72$ which is III.

Our answers are then I and III.

Solve the triangle

Law_of_cosines__sss_

$A = 115^{\circ}, B = 22^{\circ}, C=43^{\circ}$

CORRECT

$A = 100^{\circ}, B = 31^{\circ}, C=49^{\circ}$

$A = 95^{\circ}, B = 23^{\circ}, C=62^{\circ}$

None of the other answers

$A = 99^{\circ}, B = 21^{\circ}, C=54^{\circ}$

Explanation

Since we are given all 3 sides, we can use the Law of Cosines in the angle form:

$cos(A) = \frac{-a^2+b^2+c^2}{2bc}$

$cos(B) = \frac{a^2-b^2+c^2}{2ac}$

$cos(C) = \frac{a^2+b^2-c^2}{2ab}$

Let's start by finding angle A:

$cos(A) = \frac{-(12)^2+5^2+9^2}{2\cdot5\cdot9} = \frac{-38}{90} = -0.4\overline{22}$

$A = cos^{-1}(-0.4\overline{22}) = 115^{\circ}$

Now let's solve for B:

$cos(B) = \frac{12^2-(5)^2+9^2}{2\cdot12\cdot9} = \frac{200}{216} = 0.\overline{925}$

$B = cos^{-1}(0.\overline{925}) = 22^{\circ}$

We can solve for C the same way, but since we now have A and B, we can use our knowledge that all interior angles of a triangle must add up to 180 to find C.

$C = 180^{\circ} - 115^{\circ} - 22^{\circ} = 43^{\circ}$

Find the length of the missing side, .

CORRECT

Explanation

First, use the Law of Sines to find the measurement of angle

$\frac{9}{\sin 35}=\frac{12}{\sin B}$

$9\sin B = 6.88$

$\sin B=0.76$

Recall that all the angles in a triangle need to add up to degrees.

Now, use the Law of Sines again to find the length of .

$\frac{d}{\sin 95.54}=\frac{9}{\sin 35}$

$d \sin 35= 8.96$

Lc1c

What is the measurement of side using the Law of Cosines? Round to the nearest tenth.

CORRECT

Explanation

The Law of Cosines for side is,

$a^2=b^2+c^2-2bc\cdot cosA$ .

Plugging in the information we know, the formula is,

Then take the square of both sides: .

Finally, round to the appropriate units: .

Pcq2

Solve for c using Law of Sines, given:

Round to the nearest tenth.

CORRECT

None of these answers are correct.

Explanation

Law of Sines

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

Therefore...

$\frac{\sin 70}{6.5}=\frac{\sin 50}{c}$

$\sin 70 = \frac{6.5 \cdot \sin 50}{c}$

$c= \frac{6.5 \cdot \sin 50}{\sin 70}$

After rounding...

Find the length of the missing side, .

CORRECT

Explanation

First, use the Law of Sines to find the measurement of angle

$\frac{6}{\sin34}=\frac{9}{\sin B}$

$6 \sin B = 5.03$

$\sin B=0.84$

Recall that all the angles in a triangle need to add up to degrees.

Now, use the Law of Sines again to find the length of .

$\frac{d}{\sin 88.86}=\frac{6}{\sin 34}$

$d \sin 34=5.9988$

Given three sides of the triangle below, determine the angles $\theta$ , $\phi$ , and $\beta$ in degrees.

Sss

$\theta \approx 75.5^\circ , \phi \approx 57.9^\circ , \beta \approx 46.6^\circ$

CORRECT

$\theta \approx 58.1^\circ , \phi \approx 47.0^\circ , \beta \approx 74.9^\circ$

$\theta \approx 47.0^\circ , \phi \approx 74.9^\circ , \beta \approx 58.1^\circ$

$\theta \approx 45.0^\circ , \phi \approx 65.0^\circ , \beta \approx 70.0^\circ$

Explanation

We are only given sides, so we must use the Law of Cosines. The equation for the Law of Cosines is

$a^2=b^2+c^2-2bc\cos{A}$ ,

where , and are the sides of a triangle and the angle is opposite the side .

We have three known sides and three unknown angles, so we must write the Law three times, where each equation lets us solve for a different angle.

To solve for angle $\theta$ , we write

$16^2=12^2+14^2-2(12)(14) \cos{\theta}$ and solve for $\theta$ using the inverse cosine function $\arccos$ on a calculator to get

$\theta \approx 75.5$ .

Similarly, for angle $\phi$ ,

$14^2=12^2+16^2-2(12)(16) \cos{\phi}$

$\phi \approx 57.9$

and for $\beta$ ,

$12^2=14^2+16^2-2(14)(16) \cos{\beta}$

and $\beta \approx 46.6$

Triangle_700

Given and $B=36^\circ$ , what is the measurement of to the nearest degree?

$82^\circ$

CORRECT

$82.15^\circ$

$82.2^\circ$

$83^\circ$

Explanation

Using the information we have, we can solve for $\angle C$ :

$\frac{sinB}{b}=\frac{sinC}{c}$ .

Plugging in what we know, we have:

$\frac{sin(36)}{4}=\frac{sinC}{6}$ .

Then, solve for :

$6\cdot\frac{sin(36)}{4}=sinC$ .

Simplify, then solve for : which means $sin^{-1}(.8816778784)=C$ .

Therefore, after rounding to the nearest degree, $C=62^\circ$ .

To solve for $\angle A$ , subtract $\angle B$ and $\angle C$ from $180^\circ$ : $180^\circ-36^\circ-62^\circ=82^\circ$ .

Therefore, $\angle A = 82^\circ$ .

Use the Law of Sines to solve for the specified variable.

Pcq2

Solve for . Round to the nearest tenth.

CORRECT

None of these answers are correct.

Explanation

Law of Sines

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

Therefore...

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

$\frac{\sin 60}{6}= \frac{\sin 70}{b}$

$\sin 60 = \frac{6 \cdot \sin 70}{b}$

$b=\frac{6 \cdot \sin 70}{\sin 60}$

After rounding...

Find the length of the missing side,

CORRECT

Explanation

First, use the Law of Sines to find the measurement of angle

$\frac{5}{\sin 50}=\frac{3}{\sin C}$

$5\sin C = 2.30$

$\sin C=0.46$

Recall that all the angles in a triangle need to add up to degrees.

Now, use the Law of Sines again to find the length of .

$\frac{d}{\sin 102.61}=\frac{5}{\sin 50}$

$d \sin 50= 4.88$

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