Math Problem Statement

Work out the size of angle θ. Give your answer to 1 decimal place.

Solution

To solve for the size of angle θ\theta, we can use the Law of Cosines or trigonometry. Since the figure appears to be a triangle, we can apply the sine rule or cosine rule if we have the correct angles or sides.

From the image, we have the following information:

  • The sides are 42.8 cm and 37.5 cm.
  • We are looking for the angle θ\theta between these sides.

We can use the Cosine Rule formula to calculate θ\theta:

cos(θ)=b2+c2a22bc\cos(\theta) = \frac{b^2 + c^2 - a^2}{2bc}

Where:

  • a=42.8cma = 42.8 \, \text{cm}
  • b=37.5cmb = 37.5 \, \text{cm}
  • c=37.5cmc = 37.5 \, \text{cm} (since the triangle is isosceles)

Let's now calculate the angle θ\theta. math

Given values

a = 42.8 # cm b = 37.5 # cm c = 37.5 # cm

Applying Cosine Rule: cos(theta) = (b^2 + c^2 - a^2) / (2bc)

cos_theta = (b2 + c2 - a**2) / (2 * b * c)

Calculating theta in radians and converting to degrees

theta_radians = math.acos(cos_theta) theta_degrees = math.degrees(theta_radians)

Rounding to 1 decimal place

theta_degrees_rounded = round(theta_degrees, 1) theta_degrees_roundedThe size of angle θ\theta is approximately 69.6° to 1 decimal place.

Do you need any further explanation or have any more questions on this? Here are some related questions you might find interesting:

  1. How do the sine and cosine rules differ in solving triangles?
  2. How would the angle change if the triangle were not isosceles?
  3. What is the Law of Sines, and when is it useful?
  4. Can this problem be solved using basic trigonometric ratios?
  5. How does the angle θ\theta change if the sides are altered?

Tip: When using the Law of Cosines or Law of Sines, always ensure you know the type of triangle (e.g., isosceles, scalene) to select the right approach for solving.

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Math Problem Analysis

Mathematical Concepts

Trigonometry
Cosine Rule
Triangle Geometry

Formulas

cos(θ) = (b² + c² - a²) / (2bc)

Theorems

Cosine Rule

Suitable Grade Level

Grades 9-11