Problem 4 - Algebra 2

Question

In \(\triangle P Q R, \angle R\) is a right angle and cot \(P=\frac{5}{12} .\) Draw a diagram. Find the values of the other five trigonometric functions of \(\angle P\) in fraction and in decimal form.

Step-by-Step Solution

Answer

The six trigonometric ratios for angle \(\angle P\) in their fractional form are: \(\sin P = \frac{12}{13}\), \(\cos P = \frac{5}{13}\), \(\tan P = \frac{12}{5}\), \(\csc P = \frac{13}{12}\), \(\sec P = \frac{13}{5}\), and \(cot P = \frac{5}{12}\). In decimal form, these ratios are approximately: \(\sin P \approx 0.923\), \(\cos P \approx 0.385\), \(\tan P \approx 2.4\), \(\csc P \approx 1.083\), \(\sec P \approx 2.6\), \(\cot P \approx 0.417\).

Step 1: Draw the Triangle

Draw a right-angled triangle \(PQR\) such that \(\angle R\) is the right angle and \(\angle P\) has a cotangent ratio of \(\frac{5}{12}\). Label the sides of the triangle from the perspective of \(\angle P\) such that the adjacent side is 5 units, the opposite side is 12 units, and use the Pythagorean theorem to find that the hypotenuse \(\sqrt{(12^2 + 5^2)}\) is 13 units.

Step 2: Calculate the Six Trigonometric Ratios

Define the six trigonometric ratios for \(\angle P\) based on the side lengths calculated in Step 1.\ \n1. Sine (\(\sin\)) of angle P = \(\frac{Opposite}{Hypotenuse} = \frac{12}{13}\) \ \n2. Cosine (\(\cos\)) of angle P = \(\frac{Adjacent}{Hypotenuse} = \frac{5}{13}\) \ \n3. Tangent (\(\tan\)) of angle P = \(\frac{Opposite}{Adjacent} = \frac{12}{5}\) \ \n4. Cosecant (\(\csc\)) of angle P = \(\frac{1}{\sin P} = \frac{13}{12}\) \ \n5. Secant (\(\sec\)) of angle P = \(\frac{1}{\cos P} = \frac{13}{5}\) \ \n6. Cotangent (\(\cot\)) of angle P = \(\frac{1}{\tan P} = \frac{5}{12}\)

Step 3: Convert to Decimal Form

Convert each of the six trigonometric ratios calculated in Step 2 to their equivalent decimal form.\ \n1. \(\sin P \approx 0.923\) \ \n2. \(\cos P \approx 0.385\) \ \n3. \(\tan P \approx 2.4\) \ \n4. \(\csc P \approx 1.083\) \ \n5. \(\sec P \approx 2.6\) \ \n6. \(\cot P \approx 0.417\)

Key Concepts

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. This angle is called the right angle. In any right triangle, the three sides are referred to as the adjacent side, the opposite side, and the hypotenuse. The hypotenuse is always the longest side and is directly opposite the right angle.

• Adjacent Side: This side is next to the angle you are considering, but not the hypotenuse.
• Opposite Side: This is the side opposite to the angle you are considering.
• Hypotenuse: This side is the diagonal across from the right angle.

Understanding the distinction between the sides and angles is crucial for applying trigonometric ratios and the Pythagorean Theorem accurately.

Trigonometric Ratios

Trigonometric ratios are relationships between the sides of a right triangle with respect to its angles. The most commonly used trigonometric ratios are sine, cosine, and tangent, which can help determine unknown side lengths when one angle and one side length are known.

• Sine (\( ext{sin}\)): The ratio of the length of the opposite side to the hypotenuse.
• Cosine (\( ext{cos}\)): The ratio of the length of the adjacent side to the hypotenuse.
• Tangent (\( ext{tan}\)): The ratio of the length of the opposite side to the adjacent side.

Additionally, the reciprocals of these ratios give us cosecant (\( ext{csc}\)), secant (\( ext{sec}\)), and cotangent (\( ext{cot}\)). Each of these functions provides a different way to relate the dimensions of the triangle, offering flexibility for solving various trigonometric problems.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that applies specifically to right triangles. It establishes a relationship between the lengths of the sides of these triangles. The theorem states:\[ c^2 = a^2 + b^2 \]Where \(c\) represents the length of the hypotenuse, and \(a\) and \(b\) represent the lengths of the other two sides. This means that if you know the lengths of two sides of a right triangle, you can always calculate the third side.

• In the example given, the sides are 12 units (opposite), 5 units (adjacent), and the hypotenuse was found using this theorem to be 13 units.

Remember, this theorem only applies to right triangles, so always verify that you have a right triangle before applying it.

Angle P

In trigonometry, angles are typically measured in degrees or radians. In the context of a right triangle like \( \triangle PQR \), angle \( P \) plays a central role in calculating trigonometric ratios and using them to find unknown sides.

• Understanding angle \( P \) helps define the reference angle for determining the adjacent and opposite sides.
• Since we know angle \( R \) is 90 degrees, angle \( P \) is computed by considering the trigonometric ratios like cotangent, which is given as \( \cot P = \frac{5}{12} \).

By calculating these ratios, you can easily find all other trigonometric functions for angle \( P \), facilitating a comprehensive understanding of the triangle's geometry.