Problem 22 - Algebra and Trigonometry

Question

Sketch a triangle that has acute angle \(\theta,\) and find the other five trigonometric ratios of \(\theta\) . $$ \tan \theta=\sqrt{3} $$

Step-by-Step Solution

Answer

The trigonometric ratios are: \( \sin \theta = \frac{\sqrt{3}}{2} \), \( \cos \theta = \frac{1}{2} \), \( \sec \theta = 2 \), \( \csc \theta = \frac{2\sqrt{3}}{3} \), \( \cot \theta = \frac{\sqrt{3}}{3} \).

Step 1: Understand Given Information

We are given that \( \tan \theta = \sqrt{3} \). This means that in a right triangle, the ratio of the opposite side to the adjacent side for angle \( \theta \) is \( \sqrt{3} \).

Step 2: Identify Triangle Sides

To find the other trigonometric ratios, consider a right triangle where the opposite side \( O \) is \( \sqrt{3} \), and the adjacent side \( A \) is \( 1 \). This satisfies \( \tan \theta = \frac{O}{A} = \sqrt{3} \).

Step 3: Calculate Hypotenuse Using Pythagoras

The hypotenuse \( H \) of the triangle can be found using the Pythagorean theorem: \( H = \sqrt{(O)^2 + (A)^2} = \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \).

Step 4: Compute Sine of Theta

The sine of \( \theta \) is the ratio of the opposite side to the hypotenuse: \( \sin \theta = \frac{\sqrt{3}}{2} \).

Step 5: Compute Cosine of Theta

The cosine of \( \theta \) is the ratio of the adjacent side to the hypotenuse: \( \cos \theta = \frac{1}{2} \).

Step 6: Compute Secant of Theta

The secant of \( \theta \) is the reciprocal of \( \cos \theta \): \( \sec \theta = \frac{1}{\cos \theta} = 2 \).

Step 7: Compute Cosecant of Theta

The cosecant of \( \theta \) is the reciprocal of \( \sin \theta \): \( \csc \theta = \frac{1}{\sin \theta} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \).

Step 8: Compute Cotangent of Theta

The cotangent of \( \theta \) is the reciprocal of \( \tan \theta \): \( \cot \theta = \frac{1}{\tan \theta} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \).

Step 9: Sketch the Triangle

Draw a right triangle with one angle as \( \theta \). Label the opposite side \( \sqrt{3} \), the adjacent side \( 1 \), and the hypotenuse \( 2 \). This triangle represents the given \( \tan \theta \). Ensure all angles and sides are appropriately marked.

Key Concepts

Right Triangle

A right triangle is a fundamental shape in trigonometry, characterized by having one angle that is exactly 90 degrees. This 90-degree angle, also known as the right angle, creates a unique relationship between the three sides of the triangle: the hypotenuse, the opposite side, and the adjacent side. Each side plays a crucial role in calculating the trigonometric ratios.

• Hypotenuse: This is the longest side and is opposite to the right angle.
• Opposite Side: This side is opposite to the angle of interest, in this case, angle \( \theta \).
• Adjacent Side: This side forms the angle \( \theta \) with the hypotenuse.

When solving trigonometry problems, it's crucial to keep these side definitions in mind, as they serve as the basis for understanding trigonometric ratios.

Pythagorean Theorem

The Pythagorean theorem is a critical concept in geometry and trigonometry. It relates the lengths of the sides in a right triangle. The theorem states that the square of the length of the hypotenuse (\( c \)) is equal to the sum of the squares of the lengths of the other two sides (\( a \) and \( b \)). Mathematically, it can be written as:

• \( a^2 + b^2 = c^2 \)

This relationship is used to find the length of any missing side when two sides are known. For instance, in our exercise:

• The opposite side \( O = \sqrt{3} \)
• The adjacent side \( A = 1 \)
• Using \( H = \sqrt{(O)^2 + (A)^2} = \sqrt{4} = 2 \)

This allows us to compute one of the most vital sides, the hypotenuse, which is necessary for calculating other trigonometric ratios.

Trigonometry

Trigonometry revolves around the relationships between the angles and sides of triangles, especially right triangles. These relationships are captured by the trigonometric ratios:

• \( \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \)
• \( \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)
• \( \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} \)

In our scenario, with \( \tan \theta = \sqrt{3} \), we calculated:

• \( \sin \theta = \frac{\sqrt{3}}{2} \)
• \( \cos \theta = \frac{1}{2} \)

Additionally, there are the reciprocal ratios that offer more insight:

• \( \sec \theta = 2 \)
• \( \csc \theta = \frac{2\sqrt{3}}{3} \)
• \( \cot \theta = \frac{\sqrt{3}}{3} \)

These ratios enable us to delve deeper into understanding the properties and behaviors of right triangles.

Acute Angles

Acute angles are those that measure less than 90 degrees. In a right triangle, apart from the right angle, the other two angles are acute. These angles are essential in trigonometry, as they define the ratios that characterize the triangle's geometry.

• \( \theta \): This angle is one of the acute angles, with a tangent of \( \sqrt{3} \), leading us to identify it as 60 degrees since \( \tan 60^\circ = \sqrt{3} \).
• The other angle is then given as \( 30^\circ \), by the triangle's angle sum property where all angles add up to \( 180^\circ \) (i.e., \( 90^\circ + 60^\circ + 30^\circ = 180^\circ \)).

Recognizing acute angles helps in understanding various trigonometric functions, as specific angle values often associate with particular trigonometric ratios, aiding in quick problem solving and verification.