Triangle Analysis Using Trigonometric Rules

00:00 8 marks

In triangle ABC, side AB (c) is 8 cm, side AC (b) is 10 cm, and the angle at vertex A is 40°. Use these measurements to solve the following:

Part a

4 minutes 3 marks

Question

Calculate the length of side BC (denoted as a) using the cosine rule, given that in triangle ABC, side AC (b) = 10 cm, side AB (c) = 8 cm, and the included angle A = 40°.

Recall the cosine rule: \(a^2 = b^2 + c^2 – 2bc \cos A\).

Hint Hint

Hint

Apply the cosine rule formula \(a^2 = b^2 + c^2 – 2bc \cos A\) by substituting \(b=10\), \(c=8\), and \(A=40^\circ\). Ensure you take the square root to find side a.

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Complete Answer

The length of side a is \(a = \sqrt{10^2 + 8^2 – 2\times10\times8\cos(40^\circ)} = \sqrt{164 – 160\cos(40^\circ)}\) cm.

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Part b

3 minutes 2 marks

Question

Determine the measure of angle B in triangle ABC using the sine rule. Use the value of side a obtained in part (a), and given side AC (b) = 10 cm and angle A = 40°.

Recall the sine rule: \(\frac{\sin A}{a} = \frac{\sin B}{b}\).

Hint Hint

Hint

Rearrange the sine rule to solve for \(\sin B\), then use the inverse sine function to find angle B. Consider the possible range of angles and check if your answer is reasonable within a triangle.

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Complete Answer

Angle B is given by \(B = \arcsin\left(\frac{10\sin(40^\circ)}{\sqrt{164 – 160\cos(40^\circ)}}\right)\).

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Part c

4 minutes 3 marks

Question

Calculate the area of triangle ABC using a trigonometric formula that involves two sides and the included angle. Use the given side lengths AC (b) = 10 cm, AB (c) = 8 cm, and the included angle A = 40°.

Use the formula: \(\text{Area} = \frac{1}{2} \times b \times c \times \sin A\). State your answer with appropriate units.

Hint Hint

Hint

Substitute the given side lengths b, c, and angle A into the area formula \(\text{Area} = \frac{1}{2}bc\sin A\). Ensure you include the correct units for area in your final answer.

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Complete Answer

The area of triangle ABC is \(\text{Area} = 40\sin(40^\circ)\) cm\(^2\).

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