Question

In Problems 33-44, solve each triangle. $$ B=20^{\circ}, C=75^{\circ}, b=5 $$

Short answer

A = 85^{\circ}, a \approx 14.62, c \approx 14.33

Step-by-step solution

- Find Angle A

Use the fact that the sum of the angles in a triangle is always 180 degrees. Given that angle B is 20 degrees and angle C is 75 degrees, calculate angle A using the formula:\( A = 180^{\text{\circ}} - B - C \)Substitute the given values:\( A = 180^{\text{\circ}} - 20^{\text{\circ}} - 75^{\text{\circ}} = 85^{\text{\circ}} \)

- Use the Law of Sines

The Law of Sines states that \( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \). We can use this to find the unknown sides a and c.First, find side a:\( \frac{a}{\sin A} = \frac{b}{\sin B} \)Rearrange to solve for a:\( a = b \cdot \frac{\sin A}{\sin B} \)Substitute the known values:\( a = 5 \cdot \frac{\sin 85^{\text{\circ}}}{\sin 20^{\text{\circ}}} \)

- Calculate Side a

Calculate the numerical value of side a using a calculator for the trigonometric functions:\( a = 5 \cdot \frac{\sin 85^{\text{\circ}}}{\sin 20^{\text{\circ}}} \approx 14.62 \)

- Calculate Side c

Finally, find side c using the Law of Sines:\( \frac{c}{\sin C} = \frac{b}{\sin B} \)Rearrange to solve for c:\( c = b \cdot \frac{\sin C}{\sin B} \)Substitute the known values:\( c = 5 \cdot \frac{\sin 75^{\text{\circ}}}{\sin 20^{\text{\circ}}} \approx 14.33 \)

Key concepts

triangle solving

Solving a triangle means finding all the unknown sides and angles. Given some initial information, such as two angles and one side, we use mathematical principles and trigonometric identities to solve for the unknowns. Here, we have the angles B and C, and side b. The steps involved include:

- Finding the unknown angle using the angle sum property of a triangle.
- Using the Law of Sines to determine the missing sides.

This step-by-step approach ensures you accurately determine all unknown values, using the given data efficiently.

Law of Sines

The Law of Sines is crucial for solving triangles when certain elements are known. It states that the ratio of the length of a side of a triangle to the sine of its opposite angle is constant. Mathematically, the Law of Sines can be written as:

\[ \frac{a}{\text{sin} A} = \frac{b}{\text{sin} B} = \frac{c}{\text{sin} C} \]

This relationship allows us to find unknown sides or angles if we have enough initial information. For instance, in our exercise, we used:

\[ a = b \times \frac{\text{sin} A}{\text{sin} B} \] and
\[ c = b \times \frac{\text{sin} C}{\text{sin} B} \]

This made solving for sides \( a \) and \( c \) straightforward once we knew angles A, B, and C.

angle sum property

One of the fundamental properties of a triangle is that the sum of its internal angles always equals 180 degrees. This is known as the angle sum property. For example, when two angles of a triangle are known, the third angle can be easily found using this property. Mathematically, this can be expressed as:

\[ A + B + C = 180^{\text{\circ}} \]

In our example, given angles B and C, we found angle A by calculating:

\[ A = 180^{\text{\circ}} - B - C \]

This property is not just a rule but a fundamental characteristic that helps us bridge known and unknown quantities in triangle solving.