Problem 21 - Algebra and Trigonometry

Question

Sketch a triangle that has acute angle \(\theta,\) and find the other five trigonometric ratios of \(\theta\). $$\sec \theta=\frac{7}{2}$$

Step-by-Step Solution

Answer

\(\sin \theta = \frac{3\sqrt{5}}{7}\), \(\tan \theta = \frac{3\sqrt{5}}{2}\), \(\csc \theta = \frac{7\sqrt{5}}{15}\), \(\cot \theta = \frac{2\sqrt{5}}{15}\).

Step 1: Understand the Given Information

We are given the trigonometric value \(\sec \theta = \frac{7}{2}\). Recall that \(\sec \theta\) is the reciprocal of \(\cos \theta\), so \(\cos \theta = \frac{2}{7}\).

Step 2: Determine Possible Values of \(\theta\)

Since \( \theta \) is an acute angle, it must be between \(0^{\circ}\) and \(90^{\circ}\). This ensures that \(\cos \theta\) is positive, which it is (\(\frac{2}{7}\)).

Step 3: Construct a Right Triangle

Sketch a right triangle where angle \(\theta\) is one of the acute angles. Label the adjacent side (to \(\theta\)) as 2 and the hypotenuse as 7, based on \(\cos \theta = \frac{2}{7}\).

Step 4: Use Pythagorean Theorem to Find the Opposite Side

The Pythagorean theorem states: \[ a^2 + b^2 = c^2 \] where \(a\) and \(b\) are the lengths of the legs, and \(c\) is the hypotenuse. We have: \[ 2^2 + b^2 = 7^2 \] \[ 4 + b^2 = 49 \] \[ b^2 = 45 \] \[ b = \sqrt{45} = 3\sqrt{5} \]. So the opposite side is \(3\sqrt{5}\).

Step 5: Calculate the Trigonometric Ratios

Now calculate the other trigonometric ratios using the known sides of the triangle: \(\sin \theta = \frac{3\sqrt{5}}{7}\), \(\tan \theta = \frac{3\sqrt{5}}{2}\), \(\csc \theta = \frac{7}{3\sqrt{5}}\), and \(\cot \theta = \frac{2}{3\sqrt{5}}\).

Step 6: Simplify the Ratios

For \(\csc \theta\) and \(\cot \theta\), we should simplify by rationalizing the denominators. \(\csc \theta = \frac{7\sqrt{5}}{15}\) and \(\cot \theta = \frac{2\sqrt{5}}{15}\).

Key Concepts

Acute Angles

In geometry, an acute angle is an angle that measures between 0° and 90°. These angles are less than the right angle, making them sharp and narrow. Understanding acute angles is essential in trigonometry as they frequently appear in problems involving triangles and trigonometric ratios.

Trigonometric functions such as sine, cosine, and tangent are defined primarily on acute angles when dealing with right triangles. This is because each of the acute angles in a right triangle has significant mathematical relationships with the other two sides. These relationships help calculate distances or unknown angles in various applications ranging from physics to architecture.

When working with trigonometric ratios, it's important to remember that the sine, cosine, and tangent of an acute angle are always positive. This positivity is crucial when determining which formulas to use and what the expected outcomes should be when solving trigonometric equations.

Right Triangle Trigonometry

Right triangle trigonometry involves studying the relationships between the sides and angles of right triangles. A right triangle is one with one 90° angle, and the remaining angles must be acute. Right triangle trigonometry forms the basis of understanding larger concepts in mathematics such as the unit circle and trigonometric identities.

The primary trigonometric ratios—sine (\(\sin\)), cosine (\(\cos\)), and tangent (\(\tan\))—are functions of an angle that relate the sizes of the sides of a right triangle. They are defined as follows:

• The sine of angle \(\theta\), \(\sin \theta\), is the ratio of the length of the opposite side to the length of the hypotenuse.
• The cosine of angle \(\theta\), \(\cos \theta\), is the ratio of the length of the adjacent side to the length of the hypotenuse.
• The tangent of angle \(\theta\), \(\tan \theta\), is the ratio of the length of the opposite side to the length of the adjacent side.

Additionally, there are reciprocal functions: cosecant (\(\csc\)), secant (\(\sec\)), and cotangent (\(\cot\)). Each explores the inverse relationship of the primary functions and provides additional avenues for problem-solving in right triangles. Understanding these ratios is key for solving equations and finding missing elements in triangle problems.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in mathematics, deeply connected to right triangles. It states that, in a right triangle, the square of the length of the hypotenuse (the side opposite the right angle) is equal to the sum of the squares of the lengths of the other two sides. It is expressed as: \[a^2 + b^2 = c^2\]where \(a\) and \(b\) are the triangle's legs, and \(c\) is the hypotenuse.

This theorem is pivotal not only in geometry but also in trigonometry, where it helps derive certain trigonometric identities and solve problems involving right triangles. For example, in the context of our exercise, knowing \(\sec \theta\) allowed us to determine the two sides of the triangle, and the Pythagorean Theorem then helped calculate the unknown side.

Understanding and applying the Pythagorean Theorem enables students to solve various real-world problems, such as calculating distances, angles, or even constructing models in physics and engineering. It is one of the first powerful mathematical tools available for breaking down complex geometrical problems into simpler components.