Question

If the sides of a triangle are \(a=2, b=2,\) and \(c=3,\) find the measures of the three angles. Round to the nearest tenth.

Short answer

Angles are approximately 41.4°, 41.4°, and 97.2°.

Step-by-step solution

- Use the Law of Cosines to Find Angle A

The Law of Cosines states: \[ c^2 = a^2 + b^2 - 2ab \cos(C) \] To find angle A, we rearrange for \( \cos(A) \): \[ a^2 = b^2 + c^2 - 2bc \cos(A) \] Substitute the given values: \[ 2^2 = 2^2 + 3^2 - 2(2)(3) \cos(A) \] Simplify: \[ 4 = 4 + 9 - 12 \cos(A) \] \[ 4 = 13 - 12 \cos(A) \] Solve for \( \cos(A) \): \[ 12 \cos(A) = 9 \] \[ \cos(A) = \frac{9}{12} \] \[ \cos(A) = 0.75 \] Use the inverse cosine to find \( A \): \[ A \approx \cos^{-1}(0.75) \approx 41.4° \]

- Use the Law of Cosines to Find Angle B

Next, find angle B using the same method: \[ b^2 = a^2 + c^2 - 2ac \cos(B) \] Substitute the given values: \[ 2^2 = 2^2 + 3^2 - 2(2)(3) \cos(B) \] \[ 4 = 4 + 9 - 12 \cos(B) \] \[ 4 = 13 - 12 \cos(B) \] \[ 12 \cos(B) = 9 \] \[ \cos(B) = \frac{9}{12} \] \[ \cos(B) = 0.75 \] Use the inverse cosine to find \( B \): \[ B \approx \cos^{-1}(0.75) \approx 41.4° \]

- Calculate Angle C

The sum of the angles in a triangle is always 180°. Knowing the measures of angles A and B, we can find angle C: \[ C = 180° - A - B \] Substitute the known angles: \[ C = 180° - 41.4° - 41.4° \] Simplify: \[ C = 180° - 82.8° \] \[ C = 97.2° \]

Key concepts

trigonometry

Trigonometry is a branch of mathematics that studies the relationships between the sides and angles of triangles. It is essential for solving problems related to triangles, especially when you cannot directly measure the angles.
For instance, using trigonometric formulas like the Law of Cosines you can determine unknown angles or side lengths in any triangle, even if certain measurements are missing.
The Law of Cosines is particularly helpful because it can work with any triangle—not just right-angled ones. This makes it versatile for various applications, from geometry to physics.
Next, we'll dive into an example to show how trigonometry and the Law of Cosines can help solve for unknown angles in a triangle.

triangle angles

In any triangle, the sum of the three interior angles is always 180 degrees. This fundamental property is key to solving for unknown angles once you know at least two angles or sides.
For example, if you know two angles, you can always find the third by subtracting the sum of the known angles from 180 degrees.
But what if you only know the sides? Here, the Law of Cosines helps out. For our exercise, it's used to find angles when side lengths are known:

• Calculate \(\text{cos}(A)\) using the side lengths and the Law of Cosines: \[ a^2 = b^2 + c^2 - 2bc \text{cos}(A) \]
• Once \(\text{cos}(A)\) is found, use the inverse cosine function to determine angle A.

This step-by-step approach helps in accurately calculating the angles using known side lengths.

inverse cosine

The inverse cosine function, denoted as \(\text{cos}^{-1}\), is used to find angles when the value of the cosine is known.
In our exercise, once we determined \(\text{cos}(A)\) by rearranging the Law of Cosines formula and substituting the triangle's sides, we use \(\text{cos}^{-1}(0.75)\) to find angle A.
The inverse cosine function essentially reverses the cosine function: given a cosine value, it returns the corresponding angle. For example:

• If \(\text{cos}(A) = 0.75\), then \(\text{cos}^{-1}(0.75)\) will give us the angle A.
• In our case: \[ A \approx \text{cos}^{-1}(0.75) \approx 41.4\text{°} \]

This principle applies similarly to angle B. Finally, since the sum of angles in a triangle is 180 degrees, we calculate angle C by subtracting the sum of angles A and B from 180.