Problem 13 - Precalculus: A Concise Course

Question

The point is on the terminal side of an angle in standard position. Determine the exact values of the six trigonometric functions of the angle. $$ (5,12) $$

Step-by-Step Solution

Answer

The trigonometric functions are as follows: \(sin = \frac{12}{13}, cos = \frac{5}{13}, tan = \frac{12}{5}, csc = \frac{13}{12}, sec = \frac{13}{5}, cot = \frac{5}{12}\).

Step 1: Calculate the Hypotenuse

Find the hypotenuse of the right triangle by using the Pythagorean theorem. In this case, the adjacent side (x-coordinate) is 5 units, and the opposite side (y-coordinate) is 12 units. Therefore, the hypotenuse \( c = \sqrt{5^2 + 12^2} = \sqrt{25 + 144} = \sqrt{169} = 13 \).

Step 2: Find the Sine, Cosine, and Tangent

Calculate sine (sin), cosine (cos), and tangent (tan) of the angle. Sin is opposite over hypotenuse. Cos is adjacent over hypotenuse. Tan is opposite over adjacent. So, \( sin = \frac{12}{13}, cos = \frac{5}{13}, tan = \frac{12}{5} \).

Step 3: Find the Cosecant, Secant, and Cotangent

Compute cosecant (csc), secant (sec), and cotangent (cot) of the angle. Cosecant is the reciprocal of sine. Secant is the reciprocal of cosine. Cotangent is the reciprocal of tangent. Hence, \( csc = \frac{13}{12}, sec = \frac{13}{5}, cot = \frac{5}{12} \).

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental principle in geometry that explains the relationship between the sides of a right triangle. It's incredibly useful in trigonometry, especially when dealing with angles and their corresponding sides. The theorem 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. The equation is:

• a² + b² = c²

In the context of this problem, you are given a right triangle with one side (adjacent) being 5 and the other side (opposite) being 12. To find the hypotenuse, you apply the theorem like this:

• Calculate: 5² + 12² = 25 + 144 = 169
• Find the square root: \[ c = \sqrt{169} = 13 \]

This result means the length of the hypotenuse is 13. By knowing the lengths of all sides, you set the stage to find all trigonometric functions related to this specific angle.

sine cosine tangent

Sine, cosine, and tangent are the primary trigonometric functions that relate the angles of a right triangle to the lengths of its sides. They are fundamental in the study of triangles and circles. Here's a quick breakdown:

• Sine (\( \sin \) ): It is the ratio of the length of the opposite side to the hypotenuse. For this triangle:\[ \sin = \frac{12}{13} \]
• Cosine (\( \cos \) ): This is the ratio of the length of the adjacent side to the hypotenuse. For this triangle:\[ \cos = \frac{5}{13} \]
• Tangent (\( \tan \) ): It represents the ratio of the opposite side to the adjacent side. For this triangle:\[ \tan = \frac{12}{5} \]

These functions are not only essential for calculating angles and distances in triangles but also serve as the building blocks for more complex trigonometric concepts.

cosecant secant cotangent

While sine, cosine, and tangent might be the most familiar trigonometric functions, their reciprocals also play a significant role in understanding trigonometry fully. These are known as cosecant, secant, and cotangent.

• Cosecant (\( \csc \) ): It is the reciprocal of sine. Therefore, \( \csc = \frac{13}{12} \) for this angle's triangle.
• Secant (\( \sec \) ): The reciprocal of cosine gives you secant. Here, \( \sec = \frac{13}{5} \).
• Cotangent (\( \cot \) ): As the reciprocal of tangent, the cotangent for this angle is \( \cot = \frac{5}{12} \).

Understanding these reciprocal functions can often make solving certain types of problems easier, particularly those involving identities and transformations in advanced trigonometry.