Question

Given the right triangle with \(\mathrm{a}=3, \mathrm{~b}=4\), and \(\mathrm{c}=5\), find the values of the trigonometric functions of \(\alpha\).

Short answer

The trigonometric functions for angle α in the right triangle with sides a = 3, b = 4, and c = 5 are: \(\sin(\alpha) = \frac{3}{5}\), \(\cos(\alpha) = \frac{4}{5}\), and \(\tan(\alpha) = \frac{3}{4}\).

Step-by-step solution

Confirm it is a right triangle using the Pythagorean theorem

In any right triangle, we can use the Pythagorean theorem: \(a^2 + b^2 = c^2\). Check if the given triangle is a right triangle by substituting the given sides into the Pythagorean theorem. \\[ 3^2 + 4^2 = 9 + 16 = 25 = 5^2 \\ \] Since the equation holds true, the triangle is indeed a right triangle.

Identify angle α and sides a and b

Let's call the angle between sides a and b, the angle α. Therefore, a will always be opposite angle α and b will be adjacent to angle α. In our case, assume that side a = 3 is opposite to angle α, and side b = 4 is adjacent to angle α.

Find sine, cosine, and tangent of angle α

For a right triangle, we define three trigonometric functions of an angle α: 1. Sine (sin): \(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\) 2. Cosine (cos): \(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\) 3. Tangent (tan): \(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\) Let's find them for angle α in our triangle.

Calculate sin(α)

Using the definition of sine, we have: \(\sin(\alpha) = \frac{\text{opposite}}{\text{hypotenuse}}\). Now, substitute the appropriate values for opposite and hypotenuse from our triangle: \(\sin(\alpha) = \frac{a}{c} = \frac{3}{5}\).

Calculate cos(α)

Using the definition of cosine, we have: \(\cos(\alpha) = \frac{\text{adjacent}}{\text{hypotenuse}}\). Now, substitute the appropriate values for adjacent and hypotenuse from our triangle: \(\cos(\alpha) = \frac{b}{c} = \frac{4}{5}\).

Calculate tan(α)

Using the definition of tangent, we have: \(\tan(\alpha) = \frac{\text{opposite}}{\text{adjacent}}\). Now, substitute the appropriate values for opposite and adjacent from our triangle: \(\tan(\alpha) = \frac{a}{b} = \frac{3}{4}\). The trigonometric functions for angle α in our right triangle are: sin(α) = \(\frac{3}{5}\) cos(α) = \(\frac{4}{5}\) tan(α) = \(\frac{3}{4}\)

Key concepts

Understanding a Right Triangle

A right triangle is a type of triangle where one of its angles is exactly 90 degrees. This angle is called the right angle. The side opposite this right angle is known as the hypotenuse, which is the longest side of the triangle. The other two sides are referred to as the legs.
In our given exercise, the sides of the right triangle are labeled as \(a = 3\), \(b = 4\), and \(c = 5\). Here, \(c\), being the longest side, serves as the hypotenuse. The legs are the shorter sides, \(a\) and \(b\), which are opposite and adjacent to the angle \(\alpha\) that we are investigating.
Using the Pythagorean theorem, we confirmed that this is indeed a right triangle because the sum of the squares of the legs equals the square of the hypotenuse: \(3^2 + 4^2 = 9 + 16 = 25\), which matches \(5^2 = 25\). This method helps to ensure the triangle is right-angled before carrying out further trigonometric calculations.

Exploring the Pythagorean Theorem

The Pythagorean theorem is a fundamental concept in geometry that relates the lengths of the sides in a right triangle. It states that:

• The square of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the other two sides, or \(a^2 + b^2 = c^2\).

This theorem essentially confirms whether or not a triangle is a right triangle by examining the side lengths. By substitution, we demonstrated that the given triangle with sides \(a = 3\), \(b = 4\), and \(c = 5\) satisfied this condition, proving its status as a right triangle.
The usefulness of the Pythagorean theorem extends beyond mere confirmation. It allows us to derive unknown side lengths when any two lengths of a right triangle are known, thus serving as a vital tool in solving various geometric problems.

Sine, Cosine, and Tangent in Right Triangles

Trigonometric functions describe the relationships between the angles and sides of triangles, especially right triangles. The three most common trigonometric functions are sine, cosine, and tangent, which are particularly relevant in our exercise.

• Sine (\(\sin\)): The sine of an angle in a right triangle is the ratio of the length of the opposite side to the hypotenuse. In the exercise, \(\sin(\alpha) = \frac{a}{c} = \frac{3}{5}\).
• Cosine (\(\cos\)): The cosine of an angle is the ratio of the adjacent side to the hypotenuse. For our triangle, \(\cos(\alpha) = \frac{b}{c} = \frac{4}{5}\).
• Tangent (\(\tan\)): The tangent of an angle is the ratio of the opposite side to the adjacent side. Here, \(\tan(\alpha) = \frac{a}{b} = \frac{3}{4}\).

These functions are crucial in determining missing angle measures and side lengths in right triangles. By understanding these trigonometric ratios, we can solve various problems across fields such as physics and engineering.