Problem 265 - Precalculus

Question

For the following exercises, find the lengths of the missing sides if side \(a\) is opposite angle \(A,\) side \(b\) is opposite angle \(B,\) and side \(c\) is the hypotenuse. $$ \sin B=\frac{1}{2}, \quad a=20 $$

Step-by-Step Solution

Answer

The missing sides are \( b = 20\sqrt{3} \) and \( c = 40 \).

Step 1: Identify Known Values

Given that \( \sin B = \frac{1}{2} \) and \( a = 20 \), we need to find sides \( b \) and \( c \). Since \( \frac{1}{2} \) is a special angle sine value, we identify \( B \) as \( 30^{\circ} \), because \( \sin 30^{\circ} = \frac{1}{2} \).

Step 2: Use Trigonometric Ratios

In a right triangle, if angle \( B = 30^{\circ} \), then angle \( A = 60^{\circ} \) (because the sum of angles in a triangle is \( 180^{\circ} \) and one angle is \( 90^{\circ} \)). Therefore, we can use \( \tan \) or \( \cos \) to find the remaining side.

Step 3: Calculate with Known Side a and Angle B

Use \( \tan B = \frac{a}{b} \). For \( B = 30^{\circ} \), \( \tan 30^{\circ} = \frac{1}{\sqrt{3}} \), so \( \frac{20}{b} = \frac{1}{\sqrt{3}} \). Solve for \( b \): \[ b = 20\sqrt{3} \]

Step 4: Use the Pythagorean Theorem

Knowing \( a \) and \( b \), use the Pythagorean theorem to find \( c \). \[ c = \sqrt{a^2 + b^2} = \sqrt{20^2 + (20\sqrt{3})^2} = \sqrt{400 + 1200} = \sqrt{1600} = 40 \]

Step 5: Verify or Substitute Alternative Method

Alternatively, knowing \( \cos 30^{\circ} = \frac{\sqrt{3}}{2} \), use \( c = \frac{a}{\sin A} \). Since \( a = c \cdot \sin A \), and \( \sin A = \sin 60^{\circ} = \frac{\sqrt{3}}{2} \), calculate directly: \[ c = \frac{20}{\frac{\sqrt{3}}{2}} = 40 \] Further confirms \( c = 40 \).

Key Concepts

Right Triangles

In geometry, a right triangle is one type of triangle that contains one angle equal to 90 degrees. The side opposite this right angle is called the hypotenuse and is always the longest side of the triangle. The other two sides are referred to as the legs of the triangle. A right triangle allows us to apply various trigonometric functions and identities to determine unknown side lengths and angle measures. By knowing just one side length and one non-right angle, you can use trigonometric ratios like sine, cosine, and tangent, along with the Pythagorean theorem, to find the remaining sides. This makes right triangles extremely useful, especially in situations involving heights and distances.

Special Angles

Special angles are particular angle measures that have well-known trigonomic values. These include angles such as 30°, 45°, and 60°. In these special cases, the sine, cosine, and tangent values are exact and are often used because of their simplicity.

• For a 30° angle, the sine value is 1/2.
• The cosine of a 30° angle is \( \sqrt{3}/2 \).
• Meanwhile, the tangent of 30° is \( 1/\sqrt{3} \).

These values are incredibly handy when solving triangles, especially right triangles, because they allow for quick calculations. Recognizing when these values appear can save time and reduce the potential for error when solving math problems.

Pythagorean Theorem

The Pythagorean theorem is a fundamental principle in geometry used to calculate the relationship between the sides of a right triangle. It states that the sum of the squares of the lengths of the two legs is equal to the square of the hypotenuse. In mathematical form, this is expressed as:\[c^2 = a^2 + b^2\]where \( c \) is the hypotenuse, and \( a \) and \( b \) are the legs of the triangle. This theorem is widely used not just in theoretical problems but also in practical, real-world situations. Whenever you know any two side lengths of a right triangle, you can always find the third.

Sine Function

The sine function is one of the basic trigonometric functions. Its primary role is to find the ratio of the length of the opposite side to the hypotenuse in a right triangle. Mathematically, it is expressed as:\[\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}\]This ratio is crucial when calculating angles and side lengths in right triangles. For special angles like 30°, sine values are particularly easy to remember due to their simplicity, such as \( \sin 30^{\circ} = 1/2 \). Knowing these basic trigonometric ratios helps to solve many geometrical problems without the need for complex calculations.