Question

One function gives all six Given the following information about one trigonometric function, evaluate the other five functions. $$\cos \theta=\frac{5}{13} \text { and } 0<\theta<\pi / 2$$

Short answer

Question: Given \(\cos \theta = \frac{5}{13}\), and angle \(\theta\) lies in the first quadrant, find the other five trigonometric functions for \(\theta\). Answer: The other five trigonometric functions for \(\theta\) are: \(\sin \theta = \frac{12}{13}, \, \tan \theta = \frac{12}{5}, \, \csc \theta = \frac{13}{12}, \, \sec \theta = \frac{13}{5}, \, \cot \theta = \frac{5}{12}\).

Step-by-step solution

Identify the given function and angle

We are given that \(\cos \theta = \frac{5}{13}\) and the angle \(\theta\) lies in the first quadrant (because \(0 < \theta < \frac{\pi}{2}\)).

Determine the adjacent and hypotenuse sides

We know that cosine is the ratio of the lengths of the adjacent side to hypotenuse. In this case, we have $$\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}} = \frac{5}{13}$$ So, the adjacent side is 5 and the hypotenuse is 13.

Use Pythagorean theorem to find the opposite side

Since we have the adjacent side and the hypotenuse, we can use Pythagorean theorem $$(\text{adjacent})^2+(\text{opposite})^2=(\text{hypotenuse})^2$$ Substitute the values and solve for the opposite side: $$(5)^2+(\text{opposite})^2=(13)^2$$ $$25+(\text{opposite})^2=169$$ $$(\text{opposite})^2=169-25$$ $$(\text{opposite})^2=144$$ $$\text{opposite}=12$$

Calculate the remaining trigonometric functions

Now that we have all sides of the right triangle, we can calculate the sine, tangent, cosecant, secant, and cotangent as follows: 1. Sine function: \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} = \frac{12}{13}\) 2. Tangent function: \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}} = \frac{12}{5}\) 3. Cosecant function: \(\csc \theta = \frac{1}{\sin \theta} = \frac{13}{12}\) 4. Secant function: \(\sec \theta = \frac{1}{\cos \theta} = \frac{13}{5}\) 5. Cotangent function: \(\cot \theta = \frac{1}{\tan \theta} = \frac{5}{12}\) Hence, the other five trigonometric functions are: $$\sin \theta = \frac{12}{13}, \, \tan \theta = \frac{12}{5}, \, \csc \theta = \frac{13}{12}, \, \sec \theta = \frac{13}{5}, \, \cot \theta = \frac{5}{12}$$

Key concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry, especially when dealing with right triangles. It states that in a right-angled 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. Mathematically, if 'a' and 'b' represent the lengths of the legs and 'c' is the length of the hypotenuse, then the theorem is written as:
\[a^2 + b^2 = c^2\].
This theorem is not only critical in mathematics but also in various practical applications like construction, navigation, and even in the fields of art and design. When evaluating trigonometric functions, the Pythagorean Theorem allows us to find the missing side of a triangle when the other two sides are known, giving us a complete set of values to work with for calculating the trigonometric ratios.

Trigonometric Ratios

Trigonometric ratios are the foundation of trigonometry and are used to describe the relationships between the angles and sides of a right-angled triangle. The primary trigonometric ratios are sine (sin), cosine (cos), and tangent (tan), which are defined as the ratios of the sides of a right triangle in relation to one of its acute angles.

Here is how each ratio is defined:

• Sine (sin) is the ratio of the length of the opposite side to the length of the hypotenuse: \( \sin(\theta) = \frac{\text{opposite}}{\text{hypotenuse}} \).
• Cosine (cos) is the ratio of the length of the adjacent side to the length of the hypotenuse: \( \cos(\theta) = \frac{\text{adjacent}}{\text{hypotenuse}} \).
• Tangent (tan) is the ratio of the length of the opposite side to the length of the adjacent side: \( \tan(\theta) = \frac{\text{opposite}}{\text{adjacent}} \).

There are also three reciprocal trigonometric functions, cosecant (csc), secant (sec), and cotangent (cot), which are the reciprocals of sine, cosine, and tangent, respectively. Understanding these ratios is crucial for solving problems involving angles and measurements in various fields such as physics, engineering, and astronomy.

Right Triangle Trigonometry

Right triangle trigonometry deals with the relationships between the angles and sides of right triangles. Using the trigonometric ratios discussed earlier, we can evaluate trigonometric functions for any angle in a right triangle.

When provided with one trigonometric function and an angle, we can determine the remaining sides of a right triangle. This process often involves using the known ratio to find one side and then applying the Pythagorean Theorem to find the other missing side. With all three sides known, it's possible to calculate all six trigonometric functions for the angle in question.

Importance in Calculating Trigonometric Functions

The significance of right triangle trigonometry extends beyond pure mathematics. It is essential in real-world applications such as determining heights and distances, and it is often used in calculus and vector analysis. Learning to accurately apply right triangle trigonometry principles can help students excel not just in mathematics, but also in science and engineering where these concepts are applied routinely.