Given a = 4, c = 3 and angle B = 100 degrees, find the measure of angle C to the nearest degree.

Given a = 4, c = 3 and angle B = 100 degrees, find the measure of angle C to the nearest degree.

The Correct Answer and Explanation is:

Correct answer:
Angle C ≈ 33°

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Explanation

To find the measure of angle C in a triangle where side a = 4, side c = 3, and angle B = 100°, we can apply the Law of Sines. This law relates the sides and angles of any triangle using the formula: asin⁡A=bsin⁡B=csin⁡C\frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C}sinAa​=sinBb​=sinCc​

In this problem, we are given:

• side a = 4
• side c = 3
• angle B = 100°

We want to find angle C.

Step 1: Use the Law of Sines

We will write: asin⁡A=csin⁡C\frac{a}{\sin A} = \frac{c}{\sin C}sinAa​=sinCc​

But we do not know angle A yet. We can instead use this variation: 4sin⁡A=3sin⁡Candangle B = 100°\frac{4}{\sin A} = \frac{3}{\sin C} \quad \text{and} \quad \text{angle B = 100°}sinA4​=sinC3​andangle B = 100°

Since the sum of angles in any triangle is 180°, we can express angle A as: A=180°−B−C=80°−CA = 180° – B – C = 80° – CA=180°−B−C=80°−C

Now we substitute into the Law of Sines: 4sin⁡(80°−C)=3sin⁡C\frac{4}{\sin (80° – C)} = \frac{3}{\sin C}sin(80°−C)4​=sinC3​

This equation is difficult to solve algebraically, so instead, we try a numerical approach by guessing values of angle C and refining our guess.

Try C = 33°:

Then A = 180 – 100 – 33 = 47°

Now apply Law of Sines: asin⁡A=csin⁡C⇒4sin⁡47°≈5.478and3sin⁡33°≈5.507\frac{a}{\sin A} = \frac{c}{\sin C} \quad \Rightarrow \quad \frac{4}{\sin 47°} \approx 5.478 \quad \text{and} \quad \frac{3}{\sin 33°} \approx 5.507sinAa​=sinCc​⇒sin47°4​≈5.478andsin33°3​≈5.507

Very close match. So angle C is approximately 33°.

This method shows how we can find missing angles using known sides and one angle, applying the Law of Sines and using the angle sum property of triangles.