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Magazine Chapter 6: Problem 22
Sketch a triangle that has acute angle \(\theta,\) and find the other five trigonometric ratios of \(\theta\) . $$ \tan \theta=\sqrt{3} $$
Short Answer
Expert verified
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 by step solution
01
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 \( \sqrt{3} \).
02
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 \( \tan \theta = \frac{O}{A} = \sqrt{3} \).
03
Calculate Hypotenuse Using Pythagoras
The hypotenuse \( H \) of the triangle can be found using the Pythagorean theorem: \( H = \sqrt{(O)^2 + (A)^2} = \sqrt{(\sqrt{3})^2 + 1^2} = \sqrt{3 + 1} = \sqrt{4} = 2 \).
04
Compute Sine of Theta
The sine of \( \theta \) is the ratio of the opposite side to the hypotenuse: \( \sin \theta = \frac{\sqrt{3}}{2} \).
05
Compute Cosine of Theta
The cosine of \( \theta \) is the ratio of the adjacent side to the hypotenuse: \( \cos \theta = \frac{1}{2} \).
06
Compute Secant of Theta
The secant of \( \theta \) is the reciprocal of \( \cos \theta \): \( \sec \theta = \frac{1}{\cos \theta} = 2 \).
07
Compute Cosecant of Theta
The cosecant of \( \theta \) is the reciprocal of \( \sin \theta \): \( \csc \theta = \frac{1}{\sin \theta} = \frac{2}{\sqrt{3}} = \frac{2\sqrt{3}}{3} \).
08
Compute Cotangent of Theta
The cotangent of \( \theta \) is the reciprocal of \( \tan \theta \): \( \cot \theta = \frac{1}{\tan \theta} = \frac{1}{\sqrt{3}} = \frac{\sqrt{3}}{3} \).
09
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.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
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.
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:
- \( a^2 + b^2 = c^2 \)
- The opposite side \( O = \sqrt{3} \)
- The adjacent side \( A = 1 \)
- Using \( H = \sqrt{(O)^2 + (A)^2} = \sqrt{4} = 2 \)
Trigonometry
Trigonometry revolves around the relationships between the angles and sides of triangles, especially right triangles. These relationships are captured by the trigonometric ratios:
- \( \sin \theta = \frac{\text{Opposite}}{\text{Hypotenuse}} \)
- \( \cos \theta = \frac{\text{Adjacent}}{\text{Hypotenuse}} \)
- \( \tan \theta = \frac{\text{Opposite}}{\text{Adjacent}} \)
- \( \sin \theta = \frac{\sqrt{3}}{2} \)
- \( \cos \theta = \frac{1}{2} \)
- \( \sec \theta = 2 \)
- \( \csc \theta = \frac{2\sqrt{3}}{3} \)
- \( \cot \theta = \frac{\sqrt{3}}{3} \)
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., \( 90^\circ + 60^\circ + 30^\circ = 180^\circ \)).
