Question

In a triangle \(\mathrm{ABC}, \mathrm{a}=(1+\sqrt{3}) \mathrm{cm}, \mathrm{b}=2 \mathrm{~cm}, \angle \mathrm{C}=60^{\circ}\) then the other two angles \(\mathrm{A}\) and \(\mathrm{B}\) and the side \(\mathrm{c}\) are respectively given by (a) \(\left(75^{\circ}, 45^{\circ}\right) ; \sqrt{6}\) (b) \(\left(45^{\circ}, 75^{\circ}\right) ; \sqrt{6}\) (c) \(\left(30^{\circ}, 90^{\circ}\right) ; \sqrt{6}\) (d) \(\left(90^{\circ}, 30^{\circ}\right) ; \sqrt{2}\)

Short answer

In a triangle with sides a = 1 + sqrt(3), b = 2, and angle C = 60 degrees, we can use the Laws of Cosines and Sines to find side c and angles A and B. After calculation, we found that side c = √6, angle A = 45 degrees, and angle B = 75 degrees. The answer that matches our calculated values is (b) (45°, 75°); √6.

Step-by-step solution

Use the Law of Cosines to find side c

The Law of Cosines states that for a triangle with sides a, b, c and corresponding angles A, B, C: $$c^2 = a^2 + b^2 - 2abcos(C)$$ In our case, a = 1 + sqrt(3), b = 2cm, and C = 60 degrees. Therefore, we need to calculate the value of c^2 as $$c^2 = (1+\sqrt{3})^2 + 2^2 - 2(1+\sqrt{3})(2)cos(60^{\circ})$$ Find the value of c^2 in the equation.

Calculate the side c

After evaluating step 1, the value for c^2 is found to be 6. To find the value of side c, we need to find the square root of 6: $$c = \sqrt{6}$$ Now we have side c = √6.

Use the Law of Sines to find angle A

The Law of Sines states that in any triangle the ratio of the length of a side to the sine of its opposite angle is the same for all three sides. In our case: $$\frac{a}{sin(A)} = \frac{b}{sin(B)} = \frac{c}{sin(C)}$$ We know side a, b, and c, and angle C, so we can find angle A using: $$sin(A) = \frac{a \cdot sin(C)}{c}$$ $$sin(A)= \frac{(1+\sqrt{3})sin(60^{\circ})}{\sqrt{6}}$$ Find the value of sin(A) in the equation and use sin inverse function to find the angle A.

Calculate angle A

After evaluating step 3, the value for sin(A) is found to be sqrt(3)/(2*sqrt(2)). Therefore, we need to find the angle A using sin inverse function: $$A = sin^{-1}(\frac{\sqrt{3}}{2\sqrt{2}})$$ $$A = 45^{\circ}$$ Now we have angle A = 45 degrees.

Find angle B

In a triangle, the sum of all angles is 180 degrees. So, we can find angle B using: $$B = 180^{\circ} – A – C$$ $$B = 180^{\circ} – 45^{\circ} – 60^{\circ}$$ $$B = 75^{\circ}$$ Now we have angle B = 75 degrees.

Compare the results with the given options

We have calculated side c = √6, angle A = 45 degrees, and angle B = 75 degrees. Comparing these results with the given options, we find that: Option (a): (75°, 45°); √6 Option (b): (45°, 75°); √6 Option (c): (30°, 90°); √6 Option (d): (90°, 30°); √2 Option (b) correctly matches our calculated values. Therefore, the answer is (b) \((45^{\circ}, 75^{\circ}) ; \sqrt{6}\).

Key concepts

Law of Cosines

To solve for an unknown side in a triangle when you know two sides and the included angle, the Law of Cosines is immensely helpful.
This law relates the lengths of the sides of a triangle to the cosine of one of its angles.
The formula is given by:- \(c^2 = a^2 + b^2 - 2ab\cos(C)\)In our exercise, the triangle has sides \(a = 1 + \sqrt{3}\), \(b = 2\ \text{cm}\), and an angle \(C = 60^{\circ}\) between them. To find side \(c\), plug these values into the formula.
Calculate step-by-step:

• Solve \((1+\sqrt{3})^2\)
• Add \(2^2\)
• Calculate \(- 2(1+\sqrt{3}) \times 2 \times \cos(60^{\circ})\)

Add these up to find \(c^2 = 6\). So, \(c = \sqrt{6}\). This process ensures that we determine the correct side length using known measurements.

Law of Sines

Once we have calculated side \(c\), we can use the Law of Sines to find unknown angles in the triangle.
The Law of Sines helps when you have a side-angle pair and need to find other elements.
Expressed as:- \(\frac{a}{\sin(A)} = \frac{b}{\sin(B)} = \frac{c}{\sin(C)}\)We need to determine angle \(A\).
Using angle \(C = 60^{\circ}\) and side \(c = \sqrt{6}\), rearrange to solve for \(\sin(A)\):

• \(\sin(A) = \frac{(1+\sqrt{3}) \times \sin(60^{\circ})}{\sqrt{6}}\)

Given that \(\sin(60^{\circ}) = \frac{\sqrt{3}}{2}\), substitute and solve the expression:

• \(\sin(A) = \frac{(1+\sqrt{3}) \cdot \frac{\sqrt{3}}{2}}{\sqrt{6}}\)
• Simplify the equation to find \(\sin(A)\)

Use the inverse sine function to determine \(A = 45^{\circ}\).
The Law of Sines is vital for calculating unknown angles accurately with known sides.

Triangle Angles

In any triangle, the sum of the interior angles always equals \(180^{\circ}\). This fundamental property allows us to compute an unknown angle if we know the other two.
It's a straightforward but essential law in trigonometry.From our calculations, we know:- Angle \(A = 45^{\circ}\)- Angle \(C = 60^{\circ}\)To find angle \(B\), use the following:

• \(B = 180^{\circ} - A - C\)

Substitute the known values:

• \(B = 180^{\circ} - 45^{\circ} - 60^{\circ}\)

Simplifying this, we find \(B = 75^{\circ}\).
Understanding the sum of angles is critical for solving and verifying triangle problems.