Problem 124 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch a right triangle corresponding to the trigonometric function of the acute angle \(\theta\). Use the Pythagorean Theorem to determine the third side and then find the values of the other five trigonometric functions of \(\theta\) \(\tan \theta=2\)

Step-by-Step Solution

Answer

The hypotenuse of the right triangle is \(\sqrt{5}\), and the six trigonometric functions of \(\theta\) are \(\tan \theta = 2\), \(\sin \theta = \frac{2}{\sqrt{5}}\), \(\cos \theta = \frac{1}{\sqrt{5}}\), \(\cot \theta = \frac{1}{2}\), \(\sec \theta = \sqrt{5}\), \(\csc \theta = \frac{\sqrt{5}}{2}\).

Step 1: Sketch the Triangle

Start by sketching a right triangle. Consider angle \(\theta\) as one of the acute angles. As per the given data, \(\tan \theta = 2\), which can be expressed as the ratio of the opposite side to the adjacent side. Let's assume the opposite side as '2' and the adjacent side as '1'. Label these on the right triangle sketch.

Step 2: Applying Pythagorean theorem

Once the triangle sides are defined, we can find the third side, which is the hypotenuse. The Pythagorean theorem states that in a right triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides. Which means \(c = \sqrt{a^2 + b^2}\), where 'c' is the hypotenuse, 'a' and 'b' are the other two sides. Hence, the hypotenuse= \(\sqrt{1^2 + 2^2}\)= \(\sqrt{5}\). Now, the sides of the triangle are defined with lengths 1, 2, and \(\sqrt{5}\).

Step 3: Calculating the trigonometric functions

Now, the other five trigonometric functions can be calculated. The sine of \(\theta\) or \(\sin \theta\) equals to the ratio of the opposite side to the hypotenuse. Hence, \(\sin \theta = \frac{2}{\sqrt{5}}\). The cosine of \(\theta\) or \(\cos \theta\) equals to the ratio of the adjacent side to the hypotenuse. Hence, \(\cos \theta = \frac{1}{\sqrt{5}}\). The cotangent of \(\theta\) or \(\cot \theta\) equals to the reciprocal of the tangent of \(\theta\). Hence, \(\cot \theta = \frac{1}{2}\). The secant of \(\theta\) or \(\sec \theta\) equals to the reciprocal of the cosine of \(\theta\). Hence, \(\sec \theta = \sqrt{5}\). The cosecant of \(\theta\) or \(\csc \theta\) equals to the reciprocal of the sine of \(\theta\). Hence, \(\csc \theta = \frac{\sqrt{5}}{2}\).

Step 4: Final Result

The six trigonometric functions of \(\theta\) are hence as follows: \(\tan \theta = 2\), \(\sin \theta = \frac{2}{\sqrt{5}}\), \(\cos \theta = \frac{1}{\sqrt{5}}\), \(\cot \theta = \frac{1}{2}\), \(\sec \theta = \sqrt{5}\), \(\csc \theta = \frac{\sqrt{5}}{2}\).

Key Concepts

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry that applies to right triangles, and it plays a crucial role in trigonometry. It states that 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. This can be expressed with the formula:
\[ c^2 = a^2 + b^2 \]
where \( c \) is the hypotenuse, and \( a \) and \( b \) are the other two sides. Understanding this theorem is crucial for finding the unknown lengths of sides in a right triangle when two sides are known, which is often necessary when solving trigonometric problems.

Right Triangle

A right triangle is a type of triangle that has one angle measuring exactly 90 degrees. The longest side of a right triangle, which is opposite the right angle, is known as the hypotenuse. The other two sides are called the legs of the triangle and can be referred to as the adjacent and opposite sides, relative to the angle of interest in trigonometric calculations. The properties of right triangles are essential in trigonometry for defining and calculating the six trigonometric functions.

Tangent Function

The tangent function is one of the six fundamental trigonometric functions. In a right triangle, the tangent of an angle \( \theta \) is the ratio of the length of the opposite side to that of the adjacent side. This is represented by the equation:
\[ \tan \theta = \frac{\text{opposite}}{\text{adjacent}} \]
In the context of the exercise problem where \( \tan \theta = 2 \), you can choose convenient values for the opposite and adjacent sides to make the calculations easier. For instance, picking 2 and 1, respectively, since their ratio is 2, simplifies the process of solving for the hypotenuse and other trigonometric functions.

Sine and Cosine

The sine and cosine functions are another pair of important trigonometric functions. The sine of an angle \( \theta \) in a right triangle is the ratio of the length of the opposite side to the hypotenuse, while the cosine of \( \theta \) is the ratio of the length of the adjacent side to the hypotenuse. These relationships are expressed as:
\[ \sin \theta = \frac{\text{opposite}}{\text{hypotenuse}} \]
and
\[ \cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}} \]
Sin and cos are especially useful because they form the basis for defining the other trigonometric functions, and they are applicable in a wide array of problems involving angles and triangles in both pure mathematics and applied fields.