Problem 26 - Algebra and Trigonometry

Question

Solve each triangle. $$ B=20^{\circ}, \quad C=70^{\circ}, \quad a=1 $$

Step-by-Step Solution

Answer

Angle A = 90°, side b ≈ 0.94, side c ≈ 0.34.

Step 1: Determine Angle A

Use the fact that the sum of the angles in a triangle is always 180°. So, we can find angle A by subtracting the sum of angles B and C from 180°.a) 180° - B - Cb) 180° - 20° - 70° = 90°c) Therefore, angle A is 90°.

Step 2: Identify the Type of Triangle

Since angle A is 90°, the triangle is a right triangle. In a right triangle with one angle being 90°, trigonometric relationships can be used to find the other sides.

Step 3: Use Trigonometry to Find Side b

We can use the sine and cosine functions to find the lengths of the sides. Firstly, let's find side b using cosine. We know that \(\text{cos}(B) = \frac{adjacent}{hypotenuse} = \frac{b}{a}\)b) \cos(20^{\circ}) = \frac{b}{1}c) b = cos(20^{\circ}) Using a calculator, b ≈ 0.94.

Step 4: Use Trigonometry to Find Side c

To find side c, we use the sine function. We know that \(\text{sin}(B) = \frac{opposite}{hypotenuse} = \frac{c}{a} \)b) \sin(20^{\circ}) = \frac{c}{1}c) c = \sin(20^{\circ})Using a calculator, c ≈ 0.34.

Key Concepts

Right Triangle

A right triangle is a special type of triangle that includes a 90° angle. This means it has one right angle out of its three angles. Because of this, some unique properties and rules apply only to right triangles. Trigonometric functions, like sine and cosine, are extremely useful in right triangles. They help us find unknown sides and angles. Remember, the two sides that form the right angle are called the 'legs,' and the longest side, opposite the right angle, is called the 'hypotenuse.'.

• The hypotenuse is always opposite the right angle.
• The other two angles sum to 90°.
• Trigonometric relationships help determine unknown values.

Sine Function

The sine function is one of the primary trigonometric functions used to relate the angles to side lengths in a right triangle. It is defined for a given angle in a right triangle as the ratio of the length of the opposite side to the hypotenuse. Formally, for an angle \theta\, \text{sin}(\theta) = \frac{opposite}{hypotenuse}\.In the context of our exercise, to find side \ c\ , we knew:\(\text{sin}(B) = \frac{c}{a}\)

• We used the angle B = 20°.
• The hypotenuse a=1.
• By plugging in the values, \text{sin}(20°) = \frac{c}{1}\.
• Therefore, \textrœim(c = \text{sin}(20°)\.

This allows us to calculate c using a calculator, yielding c ≈ 0.34.

Cosine Function

The cosine function is another critical trigonometric function. It is the ratio of the length of the adjacent side to the hypotenuse for a given angle in a right triangle. Formally, for an angle \theta\, \text{cos}(\theta) = \frac{adjacent}{hypotenuse}\.To find side \ b\ in our exercise, we used:\(\text{cos}(B) = \frac{b}{a}\)

• Again, for angle B = 20°.
• With hypotenuse a=1.
• Plugging the values in, \text{cos}(20°) = \frac{b}{1}\.
• Therefore, \text{b = \text{cos}(20°)}\.
• Using a calculator, we find b ≈ 0.94.

This way, we apply cosine to solve for the unknown side length.

Triangle Sum Theorem

The Triangle Sum Theorem states that the sum of the internal angles of any triangle is always 180°. This is crucial for solving triangles, especially when we know some of the angles already.In our exercise, we were given angles B and C, but we needed to find angle A. We used:
180° - B - C = A
Plugging in the values:
180° - 20° - 70° = 90°
So, angle A = 90°. This told us the triangle is a right triangle, because one of its angles is a right angle (90°). The Triangle Sum Theorem is fundamental in trigonometry for solving unknown angles. It assures that knowing two angles and solving for the third is straightforward.