Question

Find the values of all the trigonometric functions for \(\theta\) equal \(\begin{array}{llll}\text { to }(1) & 0^{\circ}(2) & 30^{\circ}(3) & 45^{\circ}(4) & 60^{\circ}(5) & 90^{\circ}\end{array}\)

Short answer

For the given angles, we have the following trigonometric function values: \(0^\circ\): \(\sin=0\), \(\cos=1\), \(\tan=0\), \(\csc=\text{undefined}\), \(\sec=1\), \(\cot=\text{undefined}\) \(30^\circ\): \(\sin=0.5\), \(\cos=\frac{\sqrt{3}}{2}\), \(\tan=\frac{\sqrt3}{3}\), \(\csc=2\), \(\sec=\frac{2\sqrt{3}}{3}\), \(\cot=\frac{1}{\sqrt{3}}\) \(45^\circ\): \(\sin=\frac{\sqrt{2}}{2}\), \(\cos=\frac{\sqrt{2}}{2}\), \(\tan=1\), \(\csc=\sqrt{2}\), \(\sec=\sqrt{2}\), \(\cot=1\) \(60^\circ\): \(\sin=\frac{\sqrt{3}}{2}\), \(\cos=0.5\), \(\tan=\sqrt{3}\), \(\csc=\frac{2\sqrt{3}}{3}\), \(\sec=2\), \(\cot=\frac{\sqrt{3}}{3}\) \(90^\circ\): \(\sin=1\), \(\cos=0\), \(\tan=\text{undefined}\), \(\csc=1\), \(\sec=\text{undefined}\), \(\cot=0\)

Step-by-step solution

Definitions of Trigonometric Functions

For any angle \(\theta\), we have the following trigonometric functions: 1. \(\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}\) 2. \(\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}\) 3. \(\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}\) 4. \(\csc(\theta) = \frac{1}{\sin(\theta)}\) 5. \(\sec(\theta) = \frac{1}{\cos(\theta)}\) 6. \(\cot(\theta) = \frac{1}{\tan(\theta)}\)

Trigonometric Ratios for Standard Angles

Let's first recall the basic trigonometric ratios for standard angles: \(\sin(0^\circ) = 0\), \(\cos(0^\circ) = 1\), \(\tan(0^\circ)= 0\) \(\sin(30^\circ) = 0.5\), \(\cos(30^\circ) = \frac{\sqrt{3}}{2}\), \(\tan(30^\circ)= \frac{\sqrt3}{3}\) \(\sin(45^\circ) = \frac{\sqrt{2}}{2}\), \(\cos(45^\circ) = \frac{\sqrt{2}}{2}\), \(\tan(45^\circ)= 1\) \(\sin(60^\circ) = \frac{\sqrt{3}}{2}\), \(\cos(60^\circ) = 0.5\), \(\tan(60^\circ)= \sqrt{3}\) \(\sin(90^\circ) = 1\), \(\cos(90^\circ) = 0\), \(\tan(90^\circ)= \text{undefined}\)

Calculating the Other Trigonometric Functions

Now we will use the basic trigonometric ratios of standard angles to calculate other trigonometric functions: 1. For \(0^\circ\): \(\csc(0^\circ) = \text{undefined}\), \(\sec(0^\circ) = 1\), and \(\cot(0^\circ) = \text{undefined}\). 2. For \(30^\circ\): \(\csc(30^\circ) = 2\), \(\sec(30^\circ) = \frac{2}{\sqrt{3}}\) , and \(\cot(30^\circ) = \frac{1}{\sqrt{3}}\). 3. For \(45^\circ\): \(\csc(45^\circ) = \sqrt{2}\), \(\sec(45^\circ) = \sqrt{2}\), and \(\cot(45^\circ) = 1\). 4. For \(60^\circ\): \(\csc(60^\circ) = \frac{2}{\sqrt{3}}\), \(\sec(60^\circ) = 2\), and \(\cot(60^\circ) = \frac{\sqrt{3}}{3}\). 5. For \(90^\circ\): \(\csc(90^\circ) = 1\), \(\sec(90^\circ) = \text{undefined}\), and \(\cot(90^\circ) = 0\). So, we have found all the trigonometric functions values for the given angles \(0^\circ\), \(30^\circ\), \(45^\circ\), \(60^\circ\), and \(90^\circ\).

Key concepts

Standard Angles

Standard angles are commonly used angles in trigonometry that have known trigonometric values. The most often referenced standard angles are \(0^\circ, 30^\circ, 45^\circ, 60^\circ,\) and \(90^\circ\).
These angles are significant because their trigonometric functions have simple, exact values, which makes calculations more straightforward.
Understanding these values helps in solving numerous trigonometric problems efficiently.When you work with standard angles, you will likely encounter terms like sine, cosine, and tangent, which are the foundational trigonometric ratios. Knowing the standard angle values can assist in graphing functions and solving geometry problems.

Trigonometric Ratios

Trigonometric ratios are relationships between the sides of a right triangle. These ratios are particularly useful for calculating unknown sides or angles.
The primary trigonometric ratios are sine, cosine, and tangent, expressed as:

• \(\sin(\theta) = \frac{\text{opposite side}}{\text{hypotenuse}}\)
• \(\cos(\theta) = \frac{\text{adjacent side}}{\text{hypotenuse}}\)
• \(\tan(\theta) = \frac{\text{opposite side}}{\text{adjacent side}}\)

When you are given an angle measure, you can find its trigonometric values using these ratios.
For standard angles, these ratios often simplify to well-known fractions and hence are easily memorized for quick reference.

Sine, Cosine, Tangent

The trigonometric functions sine, cosine, and tangent are the core functions in trigonometry.
Each of these functions relates an angle to the ratio of two sides in a right triangle.
These functions are essential for understanding wave properties, circles, and oscillations in physics. To memorize them, remember:

• Sine represents the vertical position of a point on a unit circle, calculated by the length of the opposite side of an angle.
• Cosine signifies the horizontal position, determined by the adjacent side in the triangle.
• Tangent is the function you use to find the steepness or slope, as it is the ratio of the sine to the cosine of an angle.

Practicing these functions with standard angles helps in gaining familiarity, which aids in solving more complex problems later.

Cosecant, Secant, Cotangent

Cosecant, secant, and cotangent are the reciprocal trigonometric functions derived from sine, cosine, and tangent, respectively.
Understanding these is crucial as they provide alternative ways to handle complex trigonometric calculations.These functions are:

• \(\csc(\theta) = \frac{1}{\sin(\theta)}\) - Helpful when sine is very small, gives an infinite or large value.
• \(\sec(\theta) = \frac{1}{\cos(\theta)}\) - Useful when cosine is very small, helping to resolve calculations in vertical movements.
• \(\cot(\theta) = \frac{1}{\tan(\theta)}\) - Used when tangent gives a very large or infinite result, providing reciprocal insight into angles.

Grasping these reciprocals can enhance your problem-solving toolkit, allowing you to approach trigonometric equations from different angles.
They are especially helpful in calculus, computer graphics, and physics scenarios where standard functions alone may not suffice.