Problem 28 - Algebra 2

Question

Sketch a right triangle with \(\theta\) as the measure of one acute angle. Find the other five trigonometric ratios of \(\theta .\) \(\tan \theta=\frac{24}{7}\)

Step-by-Step Solution

Answer

The six trigonometric ratios for the given problem are \(\sin \theta = \frac{24}{25}\), \(\cos \theta = \frac{7}{25}\), \(\tan \theta = \frac{24}{7}\), \(\cot \theta = \frac{7}{24}\), \(\sec \theta = \frac{25}{7}\), \(\csc \theta = \frac{25}{24}\)

Step 1: Sketch the Triangle

Start by sketching a right triangle, \(\triangle ABC\), with the angle \(\theta\) opposite side BC, adjacent side AB, and hypotenuse AC.

Step 2: Assign Values to Sides

Since we know that \(\tan \theta = \frac{24}{7}\), where tangent of an angle is the ratio of the length of the opposite side to the adjacent side, we can assign the value 24 to the opposite side BC and 7 to the adjacent side AB. We still need to find the length of the hypotenuse AC by using Pythagoras' theorem, \[ AC = \sqrt{BC^2 + AB^2} = \sqrt{24^2 + 7^2}\]

Step 3: Calculate the Hypotenuse

Now calculate the square root of 625, which gives AC as 25.

Step 4: Find the Trigonometric Ratios

With all sides of triangle now known, let's compute the six trigonometric ratios as follows:\n \(\sin \theta = \frac {opposite}{hypotenuse} = \frac{24}{25}\) \n\(\cos \theta = \frac {adjacent}{hypotenuse} = \frac{7}{25}\) \n\(\tan \theta = \frac {opposite}{adjacent} = \frac{24}{7}\) (already given) \n\(\cot \theta = \frac {1}{\tan \theta} = \frac{7}{24}\) \n\(\sec \theta = \frac {1}{\cos \theta} = \frac{25}{7}\) \n\(\csc \theta = \frac {1}{\sin \theta} = \frac{25}{24}\)

Key Concepts

Pythagorean theorem

The Pythagorean theorem is a fundamental concept in geometry, especially useful when dealing with right triangles. It states that in a right 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 other two sides. This is expressed using the equation:
\[ c^2 = a^2 + b^2 \]where \( c \) is the hypotenuse and \( a \) and \( b \) are the other two sides of the triangle.

In the exercise above, we used the Pythagorean theorem to find the length of the hypotenuse AC of the triangle \( \triangle ABC \), given \( AB = 7 \) and \( BC = 24 \). By substituting these values into the theorem, we calculated:

• \( c^2 = 24^2 + 7^2 \)
• \( c^2 = 576 + 49 \)
• \( c^2 = 625 \)
• \( c = \sqrt{625} = 25 \)

Thus, the length of the hypotenuse is 25. This calculation is crucial in determining other trigonometric ratios of the triangle.

right triangle

A right triangle is a triangle in which one angle measures exactly 90 degrees. In our exercise, it's important because the relationships of trigonometric functions are based on right triangles. A right triangle has two legs, the sides that form the right angle, and a hypotenuse, which is the longest side.

In the context of this exercise:

• The right triangle has angle \( \theta \) placed as one of its acute angles.
• The opposite side of \( \theta \) is BC, assigned the value 24.
• The adjacent side to \( \theta \) is AB, assigned the value 7.
• The hypotenuse is AC, calculated as 25 using the Pythagorean theorem.

By identifying these components, we can proceed to calculating the trigonometric functions like sine, cosine, and tangent.

sine and cosine

Sine and cosine are fundamental trigonometric functions. They relate angles in right triangles to the lengths of the sides.
- The sine of angle \( \theta \), or \( \sin \theta \), is the ratio of the opposite side to the hypotenuse:
\[ \sin \theta = \frac {opposite}{hypotenuse} \]- The cosine of angle \( \theta \), or \( \cos \theta \), is the ratio of the adjacent side to the hypotenuse:
\[ \cos \theta = \frac {adjacent}{hypotenuse} \]For our triangle:

• \( \sin \theta = \frac {24}{25} \)
• \( \cos \theta = \frac {7}{25} \)

These relationships are instrumental in solving many problems involving right triangles, offering insight into the angle and side relationships.

tangent ratio

The tangent ratio is another basic trigonometric concept often introduced alongside sine and cosine. It is defined as the ratio of the opposite side to the adjacent side of a right triangle for a given angle.
- Formula: \( \tan \theta = \frac {opposite}{adjacent} \)
In our problem:

• \( \tan \theta = \frac{24}{7} \)

This value was given in the exercise and helped in figuring out the proportions of the triangle's sides initially. Once other side lengths were known, we could also derive the reciprocal tangent functions:

• \( \cot \theta = \frac{7}{24} \)

Understanding the tangent ratio is crucial for solving equations and finding the value of angles when working with right triangles, especially in cases where side lengths need to be deduced or verified.