Problem 6 - Precalculus

Question

Suppose that \(\triangle A B C\) is a right triangle with \(\angle C=90^{\circ}\) If \(A C=6\) and \(B C=2,\) find the following quantities. (a) \(\cos A, \sin A, \tan A\) (b) \(\sec B, \csc B, \cot B\)

Step-by-Step Solution

Answer

(a) \(\cos A = \frac{\sqrt{10}}{10}, \sin A = \frac{3\sqrt{10}}{10}, \tan A = 3\); (b) \(\sec B = \sqrt{10}, \csc B = \frac{\sqrt{10}}{3}, \cot B = \frac{1}{3}\).

Step 1: Identify the Triangle Sides

In triangle ABC, angle C is the right angle. AC is the opposite side to angle B, and BC is the adjacent side to angle A. The hypotenuse (AB) needs to be calculated.

Step 2: Use the Pythagorean Theorem

Apply the Pythagorean theorem to find the hypotenuse: \[ AB = \sqrt{AC^2 + BC^2} = \sqrt{6^2 + 2^2} = \sqrt{36 + 4} = \sqrt{40} = 2\sqrt{10} \]

Step 3: Calculate Trigonometric Ratios for Angle A

Calculate \(\cos A, \sin A, \tan A\) using known sides:1. \(\cos A = \frac{BC}{AB} = \frac{2}{2\sqrt{10}} = \frac{1}{\sqrt{10}} = \frac{\sqrt{10}}{10}\)2. \(\sin A = \frac{AC}{AB} = \frac{6}{2\sqrt{10}} = \frac{3}{\sqrt{10}} = \frac{3\sqrt{10}}{10}\)3. \(\tan A = \frac{AC}{BC} = \frac{6}{2} = 3\)

Step 4: Calculate Trigonometric Ratios for Angle B

Calculate \(\sec B, \csc B, \cot B\) using known sides:1. \(\sec B = \frac{AB}{BC} = \frac{2\sqrt{10}}{2} = \sqrt{10}\)2. \(\csc B = \frac{AB}{AC} = \frac{2\sqrt{10}}{6} = \frac{\sqrt{10}}{3}\)3. \(\cot B = \frac{BC}{AC} = \frac{2}{6} = \frac{1}{3}\)

Key Concepts

Pythagorean theorem

Understanding the Pythagorean theorem is key when working with right triangles. This classic theorem relates the lengths of the sides of a right triangle. In any right triangle, the sum of the squares of the two shorter sides (legs) is equal to the square of the longest side, known as the hypotenuse.

The formula is expressed as:

• \[ a^2 + b^2 = c^2 \]

where:

• \(a\) and \(b\) are the lengths of the legs.
• \(c\) is the hypotenuse.

This simple algebraic formula allows you to calculate any side of a right triangle if you know the lengths of the other two sides. In our original exercise, AC and BC are known, and we needed to find AB, the hypotenuse. By squaring the values and adding them up, the hypotenuse was found using square roots. Applying the Pythagorean theorem gave us \[2\sqrt{10}\] for the hypotenuse. Remember, this theorem only applies to right triangles.

trigonometric ratios

Trigonometric ratios provide a way to relate the angles and sides of a right triangle. These ratios are often used in problems involving right triangles and are fundamental in trigonometry.

The primary trigonometric ratios are:

• Cosine (\(\cos\)): the ratio of the adjacent side to the hypotenuse.
• Sine (\(\sin\)): the ratio of the opposite side to the hypotenuse.
• Tangent (\(\tan\)): the ratio of the opposite side to the adjacent side.

To solve problems involving these ratios, it’s crucial to correctly identify the sides in relation to the angle you’re working with. For \(\angle A\),

• \(\cos A\) was calculated as \(\frac{BC}{AB}\).
• \(\sin A\) was \(\frac{AC}{AB}\).
• \(\tan A\) was \(\frac{AC}{BC}\).

These values were found by substituting known side lengths into the formulas. Understanding these relationships helps solve anything from simple homework problems to complex engineering tasks.

hypotenuse calculation

Calculating the hypotenuse is a common task in many right triangle problems and can be quite straightforward with the right tools, like the Pythagorean theorem. The hypotenuse is always the side opposite the right angle and is the longest side in a right triangle.

In the original exercise, we used the Pythagorean theorem to find the hypotenuse, where we first squared the other two sides and then took the square root of their sum. This process gives you the length of the hypotenuse directly once other sides are known.

• Use the formula \[ AB = \sqrt{AC^2 + BC^2} \]

Remember that hypotenuse calculation also plays a role in determining trigonometric ratios. Many trigonometric functions, like cosine and sine, rely on the hypotenuse for their calculations. Always check if the triangle is indeed a right triangle before applying these methods.