Problem 123 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta\). Use the Pythagorean Theorem to determine the third side and then find the values of the other five trigonometric functions of \(\theta\) \(\sin \theta=\frac{5}{6}\)

Step-by-Step Solution

Answer

The theme for this exercise is understanding and applying the properties of a right triangle. The lengths of the triangle sides are found to be Opposite=5, Hypotenuse=6, and Adjacent=\(\sqrt{11}\). The other five trigonometric functions of \(\theta\) were found as: \(\cos \theta=\frac{\sqrt{11}}{6}\), \(\tan \theta=\frac{5}{\sqrt{11}}\), \(\csc \theta=\frac{6}{5}\), \(\sec \theta=\frac{6}{\sqrt{11}}\), \(\cot \theta=\frac{\sqrt{11}}{5}\).

Step 1: Sketching the Right Triangle

Sketch a right triangle with \(\theta\) as one of the acute angles. Label the side across \(\theta\) (i.e., the side opposite \(\theta\)) as 5, which is the 'opposite'. The hypotenuse (the side across the right angle) is labelled as 6.

Step 2: Find the Adjacent Side Using the Pythagorean Theorem

To find the length of the third side (the side adjacent to angle \(\theta\)), use the Pythagorean theorem, which states that in any right triangle, the square of the length of the hypotenuse equals the sum of the squares of the lengths of the other two sides. In this case, the length of the adjacent side can be calculated as \(\sqrt{6^2 - 5^2} = \sqrt{11}\).

Step 3: Finding the Other Trigonometric Functions

Now that all the sides of the triangle are known, we can use their length to evaluate the other five trigonometric functions: \(\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{\sqrt{11}}{6}\). \(\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}} = \frac{5}{\sqrt{11}}\). \(\csc \theta = \frac{1}{\sin \theta} = \frac{6}{5}\). \(\sec \theta = \frac{1}{\cos \theta} = \frac{6}{\sqrt{11}}\). \(\cot \theta = \frac{1}{\tan \theta} = \frac{\sqrt{11}}{5}\).

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in trigonometry and geometry. It is a relation in Euclidean geometry among the three sides of a right triangle. The theorem states:

\[ a^2 + b^2 = c^2\]
Where \(c\) represents the length of the hypotenuse, and \(a\) and \(b\) represent the lengths of the triangle's other two sides. To apply this to a trigonometric problem involving a right triangle, you take the given side lengths—in the example provided, the length of the opposite side (5) and the hypotenuse (6)—to solve for the missing length of the adjacent side. Using the theorem, we calculate it as \(\sqrt{6^2 - 5^2} = \sqrt{36 - 25} = \sqrt{11}\) units. Understanding and applying the Pythagorean Theorem is essential for solving many problems in mathematics, especially when dealing with right triangles.

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees, known as the right angle. The longest side across from the right angle is termed the hypotenuse. The other two sides, which form the right angle, are referred to as the legs of the triangle: one is adjacent to the acute angle of interest, and the other is opposite it. In trigonometry, these terms are extremely important because they help define the trigonometric functions.

For instance, in the exercise provided, after sketching the right triangle and labeling the sides based on the given sine value \(\sin \theta = \frac{5}{6}\), it becomes clear how the sides relate to angle \(\theta\). The 'opposite' side was labeled as 5 and the hypotenuse as 6. Identifying these sides correctly is a critical step before moving on to find the length of the 'adjacent' side using the Pythagorean Theorem and before calculating the remaining trigonometric functions.

Acute Angles in Trigonometry

Acute angles are angles that measure less than 90 degrees. In trigonometry, the acute angles of a right triangle are of special interest because they directly relate to the trigonometric functions sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\)). Other functions derived from these include cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)).

The sine of an acute angle in a right triangle is the ratio of the length of the side opposite the angle to the length of the hypotenuse. Similarly, cosine relates the adjacent side to the hypotenuse, while tangent uses the opposite over the adjacent side. Once you have the sine value, as in \(\sin \theta = \frac{5}{6}\), you can follow the problem-solving steps illustrated in the exercise to find the lengths of the other sides. This then enables you to calculate the remaining trigonometric functions as ratios involving these sides. These functions are crucial for solving various real-world and theoretical problems in science, engineering, and mathematics.