Problem 18 - Precalculus: A Concise Course

Question

Sketch a right triangle corresponding to the trigonometric function of the acute angle \(\boldsymbol{\theta}\). Use the Pythagorean Theorem to determine the third side and then find the other five trigonometric functions of \(\boldsymbol{\theta}\). $$ \sec \theta=\frac{17}{7} $$

Step-by-Step Solution

Answer

Following the steps, the values of the trigonometric functions are: \(\sin \theta = \frac{2\sqrt{60}}{17}\), \(\cos \theta = \frac{7}{17}\), \(\tan \theta = \frac{2\sqrt{60}}{7}\), \(\csc \theta = \frac{17}{2\sqrt{60}}\), \(\sec \theta = \frac{17}{7}\), and \(\cot \theta = \frac{7}{2\sqrt{60}}\).

Step 1: Identify sec \(\theta\)

From the exercise, we know that \(\sec \theta = \frac{17}{7}\). Since secant is the reciprocal of cosine, we can identify that the hypotenuse (H) is 17 and the adjacent (A) side to angle \(\theta\) is 7.

Step 2: Applying the Pythagorean Theorem

We can now use the Pythagorean theorem to find the length of the opposite (O) side. The theorem states that \(H^{2} = A^{2} + O^{2}\). Substituting \(H = 17\) and \(A = 7\) gives us \(289 = 49 + O^{2}\). Solving for \(O^{2}\), we get \(O^{2} = 240\), thus \(O = \sqrt{240} = 2\sqrt{60}\).

Step 3: Calculate the other trigonometric functions

Now that we have the measurements for all three sides, we can write the other trigonometric functions as follows: \(\sin \theta = \frac{O}{H} = \frac{2\sqrt{60}}{17}\), \(\cos \theta = \frac{A}{H} = \frac{7}{17}\), \(\tan \theta = \frac{O}{A} = \frac{2\sqrt{60}}{7}\), \(\csc \theta = \frac{H}{O} = \frac{17}{2\sqrt{60}}\), and \(\cot \theta = \frac{A}{O} = \frac{7}{2\sqrt{60}}\)

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, especially when dealing with right triangles. It establishes a relationship between the lengths of the three sides of a right triangle: the two legs and the hypotenuse. The legs are the two sides that form the right angle, while the hypotenuse is the longest side, opposite the right angle.

The theorem itself can be written as: \( c^2 = a^2 + b^2 \) where \( c \) represents the length of the hypotenuse, and \( a \) and \( b \) represent the lengths of the other two sides. When solving for a missing side, you can rearrange the equation accordingly.

For instance, if you know the lengths of the hypotenuse and one leg, as in our exercise where the hypotenuse (H) is 17 and the adjacent side (A) to angle \( \theta \) is 7, you can solve for the unknown leg (O) by rearranging the theorem to \( O^2 = c^2 - a^2 \) and then take the square root to find the length of the leg.

Secant (sec)

In trigonometry, secant (sec) is one of the six fundamental trigonometric functions. It is related to the cosine function (cos) as it is its reciprocal. This means that the secant of an angle \( \theta \) in a right triangle is the ratio of the Hypotenuse (H) to the Adjacent side (A): \( \text{sec} \, \theta = \frac{H}{A} \).

Therefore, if \( \text{sec} \, \theta \) is given, as in our exercise where \( \text{sec} \, \theta = \frac{17}{7} \), you can determine the lengths of the hypotenuse and the adjacent side. In real-life applications, the secant function can be used in various fields like engineering and physics, to calculate forces, wave patterns, and many other physical properties.

Right Triangle Trigonometry

Right triangle trigonometry is the branch of mathematics that deals with the relationships between the angles and sides of right triangles. The six trigonometric functions—sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot)—serve as the foundation of these relationships.

Each function—apart from secant, which was discussed earlier—relates a specific pair of sides of the triangle. For example; \( \text{sin} \, \theta \) is the ratio of the Opposite side to the Hypotenuse \( \frac{O}{H} \), while \( \text{cos} \, \theta \) is the ratio of the Adjacent side to the Hypotenuse \( \frac{A}{H} \).

Once you've established one trigonometric function, like secant, and calculated the lengths of the right triangle's sides, you can compute the other five trigonometric functions. As demonstrated in the solution, knowing the lengths of the three sides allows us to create a full set of trigonometric values for the angle \( \theta \) utilizing the various ratios of the sides.