Problem 26 - Algebra 2

Question

Sketch a right triangle with \(\theta\) as the measure of one acute angle. Find the other five trigonometric ratios of \(\theta .\) \(\cos \theta=\frac{7}{20}\)

Step-by-Step Solution

Answer

The other five trigonometric ratios of \(\theta\) are: \(\sin \theta = 0.999\), \(\tan \theta = 2.85\), \(\csc \theta = 1.00\), \(\sec \theta = 2.85\), and \(\cot \theta = 0.35\).

Step 1: Expression interpretation

Recognize that \(\cos \theta=\frac{7}{20}\) represents the ratio of the length of the adjacent side to the hypotenuse in a right triangle. Assume the adjacent side is 7 units and the hypotenuse is 20 units.

Step 2: Find the opposite side

Use the Pythagorean theorem to calculate the length of the opposite side. In any right triangle, the square of the length of the hypotenuse (c) is equal to the sum of the squares of the other two sides (a and b). Therefore, if b is the opposite side, then using the formula \(a^2 + b^2 = c^2\), we get \(b = \sqrt{c^2 - a^2} = \sqrt{(20)^2 - (7)^2} = \sqrt{399} = 19.97\) (rounded to two decimal places).

Step 3: Find the other trigonometric ratios

Now that all sides are known, the other trigonometric ratios can be calculated as follows: - \(\sin \theta = \frac{opposite}{hypotenuse} = \frac{19.97}{20} = 0.999\)- \(\tan \theta = \frac{opposite}{adjacent} = \frac{19.97}{7} = 2.85\) - \(\csc \theta = \frac{1}{\sin \theta} = \frac{1}{0.999} = 1.00\) - \(\sec \theta = \frac{1}{\cos \theta} = \frac{1}{\frac{7}{20}} = 2.85\) - \(\cot \theta = \frac{1}{\tan \theta} = \frac{1}{2.84} = 0.35\)

Key Concepts

Right Triangle

A right triangle is a special type of triangle that includes one angle measuring exactly 90 degrees. These triangles are pivotal in trigonometry because they possess unique properties that enable the calculation of trigonometric ratios. In a right triangle:

• The side opposite the right angle is called the hypotenuse, which is always the longest side.
• The other two sides are termed the opposite and adjacent sides, with respect to the angle of interest.

Understanding these basic elements is essential when dealing with trigonometric ratios and calculations.

Cosine

The cosine of an angle in a right triangle relates to two particular sides: the adjacent side to that angle and the hypotenuse. Mathematically, cosine is expressed as follows:
\[\cos \theta = \frac{\text{adjacent side}}{\text{hypotenuse}}\]
In our example exercise, the cosine is given by \( \cos \theta = \frac{7}{20} \). This tells us that the adjacent side is 7 units long while the hypotenuse is 20 units. Knowing one trigonometric ratio allows us to find others after determining all triangle sides.

Pythagorean Theorem

The Pythagorean Theorem is a fundamental principle in geometry for right triangles. It states that the square of the hypotenuse (c) is equal to the sum of the squares of the other two sides, often labeled as a and b. This relationship is summarized in the formula:
\[a^2 + b^2 = c^2\]
In our task, given \(a = 7\) and \(c = 20\), substituting these values into the formula helps us find the length of the opposite side: \(b = \sqrt{20^2 - 7^2} = \sqrt{399} \approx 19.97\). This allows us to use this calculated side length to figure out all trigonometric ratios.

Sine

The sine function is another trigonometric ratio, which involves the opposite side of the angle and the hypotenuse of the triangle. The sine of an angle \(\theta\) is computed as follows:
\[\sin \theta = \frac{\text{opposite side}}{\text{hypotenuse}}\]
For our particular triangle, substituting the known values, we get \(\sin \theta = \frac{19.97}{20} \approx 0.999\). The sine ratio helps us understand the relation of the angle \(\theta\) to the triangle's sides, especially useful when designing solutions or comparing angles in varied contexts.

Tangent

In trigonometry, the tangent of an angle is the ratio of the opposite to the adjacent side in a right triangle. Here's the formula for tangent:
\[\tan \theta = \frac{\text{opposite side}}{\text{adjacent side}}\]
Using the calculated side lengths for the given angle \(\theta\), the tangent value is \(\tan \theta = \frac{19.97}{7} \approx 2.85\). This ratio can be insightful for assortment of applications, such as calculating slopes or angles of elevation, especially in fields like physics and engineering, where such computations are frequently encountered.