Inverse Trigonometric Functions

Inverse Trigonometric Functions

Inverse Sine Function

The graph of y = sin x does not pass the vertical line test due to the different values of x that have the same y value.  However, if you restrict the doman to the interval -π/2 ≤ x ≤ π/2, it becomes a one-to-one function. 

Now the inverse is possible to graph.  

The domain of y = sin x (-π/2, π/2) becomes the inverse range and the range of y = sin x (-1, 1) becomes the inverse domain.  This is called y = arcsin x.

Inverse Cosine Function

The graph of y = cos x does not pass the vertical line test due to the different values of x that have the same y value.  However, if you restrict the doman to the interval 0 ≤ x ≤ π, it becomes a one-to-one function. 

Now the inverse is possible to graph.

The domain of y = cos x (0, π) becomes the inverse range and the range of y = cos x, (-1, 1), becomes the inverse domain.  This is called y = arccos x.

Inverse Tangent Function

The tangent graph is repetitive and therefore, the domain can be restricted to the interval (-π/2, π/2).  

The domain of y = tan x (-π/2, π/2) becomes the inverse range and the range of y = tan x (-∞, ∞) becomes the inverse domain.

Notice that the vertical asymptotes of the y = tan x graph become the horizontal asymptotes of the y = arctan x graph.

Sine and Cosine Graphs

The graphs of sine and cosine functions represent their outputs for all real number values of a given angle, making the domain equal to all real numbers. The parent functions appear as shown below:

 f(x) = sin x

f(x) = cos x

The full equations of the functions can be expressed as

and

respectively, where the absolute value of a determines the amplitude of the function, determines the period, c determines the phase shift, and d determines the mid-line shift.


Definitions: 
Amplitude: half the distance between the minimum and maximum values of the function. Variations in amplitude vertically stretch and compress the graph.
Period: one completed cycle of the repeating function. Increasing the value of b thus increases the frequency of periods in a given distance.
Phase Shift: Horizontal stretching and compressing.
Mid-line Shift: Vertical translation of the graph.


Cool things to note:
- The graph of f(x)=sin x is odd, as evidenced by the reflection about the origin.
- The graph of f(x)=cos x is even, which is seen in the symmetry about the x-axis.
- If one were to shift the sine graph   units to the left, this phase shift represents the co-function identity:

Examples:

Graph:

***It is important to realize that the period is and that the phase shift is 1/2. This becomes more evident once 1/2 is factored out of the parentheses. Otherwise, the amplitude is 2/3. Together, this determines the following graph (red) compared to the parent graph (blue):

Trigonometric Functions of Any Angle

The definitions of the trigonometric functions can be expanded to incorporate (or measure) any angle. Instead of adhering to the angular constraints of a right triangle, where an angle θ must be acute so that 0° < θ < 90°, these functions can be reapplied to any other angle θ that sits on a coordinate plane in standard position with the point (x,y) on its terminal side.

 

The length of the terminal side r can be found by using the following equation derived from the Pythagorean Theorem,

 

using x and y from the coordinate point as substitutes.

The definitions of the trigonometric functions of any angle are as follows:

 

 

In order to determine the sign of the trigonometric functions, one must determine what quadrant they are in.

For example, since sin θ = y/r, that means that sin θ is positive whenever y > 0, which is in quadrants I and II (r is always positive).

Reference Angles

Suppose θ is an angle in standard position. Its reference angle is the acute angle θ' (read as "theta prime") formed by the terminal side of  angle θ and the x-axis (horizontal axis). Reference angles are calculated based on what quadrant the terminal side of θ lies in.

 

Quadrant I:

 (radians and degrees)

 

 

Quadrant II:

 (radians)

 (degrees)

 

 

Quadrant III:

 (radians)

 (degrees)

 

 

Quadrant IV:

 (radians)

 (degrees)

 

Reference angles can also be used to determine the value of a trigonometric function. For instance, suppose one must find cos 480°. One can subtract 360° from 480° to get a co-terminal angle of 120°. The reference angle of 120° (which, in this case, is θ) is 60° because 120° is in Quadrant II, so to find θ' one simply subtracts 120° from 180° to get 60°. From there, one can deduce that cos 60° is negative because it is in Quadrant II (where x < 0, given that cos θ = x/r). Therefore, cos 480° can be rewritten as -cos 60° which is -1/2.

 



In other words, to evaluate a trigonometric function of any angle, one must calculate the reference angle and determine the function value of it. From there, one must figure out the appropriate sign of the function value by looking at what quadrant it is in.


Right Triangle Trigonometry

Right Triangle Trigonometry 

When we look at a right triangle,we can see that the shape is made up of three sides. These three sides each have their own distinct name, which vary depending on the location of the angle in question (We will use Theta, written as "θ"). The identity of the sides can be seen in the image below:

Where:

• H = Hypotenuse (Always opposite of the right angle)
• A = Side Adjacent to θ
• O = Side Opposite to θ

The identities of the three sides of a triangle can be used to define all of the six trigonometric functions (Sine, Cosine, Tangent, Secant, Cosecant, and Cotangent). The equations used to find the trigonometric functions using the three sides of a right triangle are as follows:

• Sin θ = O/H
• Cos θ = A/H
• Tan θ = O/A
• Csc θ = H/O
• Sec θ = H/A
• Cot θ = A/O

In addition to these equations, there are many other ways one can relate the sides and angles of a right triangle. These relationships can be defined as Trigonometric Identities, and there are many different kinds of them. Trigonometric identities are unique in the way that they are always true, no matter what the input of the equation is, meaning that the Domain is equal to all real numbers. There are three main categories of trigonometric identities used to relate the six trigonometric functions to each other:

1). Reciprocal Identities - Shows the relationship between each trig function and its reciprocal

• Sin θ = 1/Csc θ
• Cos θ = 1/Sec θ
• Tan θ = 1/Cot θ
• Csc θ = 1/Sin θ
• Sec θ = 1/Cos θ
• Cot θ = 1/Tan θ

2). Quotient Identities - Shows the relationship between two trig functions and the quotient of them                                            divided together

• Tan θ = Sin θ/Cos θ
• Cot θ = Cos θ/Sin θ

3). Pythagorean Identities - Shows the relationship between trig functions through the Pythagorean                                                  Theorem

• (Sin θ)^2 + (Cos θ)^2 = 1
• 1 + (Tan θ)^2 = (Sec θ)^2 
• 1 + (Cot θ)^2 = (Csc θ)^2

*The Second and Third identities can be derived from the First one.

These identities can be used in numerous ways throughout Trigonometry and Pre-Calculus, as well as Calculus. They can be used to condense and expand expressions, and solve for trigonometric functions when needed. Trigonometric Identities are extremely useful when proving relationships between functions, as well as deriving new proofs. 

Here is an Example of a problem where Trig Identities can be used to prove an equation:

Prove: (Csc θ)(Tan θ) = Sec θ

One can use the reciprocal identity to transform both Cosecant and Tangent into terms of Sine and Cosine.

(1/Sin θ)(Sin θ/Cos θ) = Sec θ

Then, one can cancel out the Sin θ's and continue to simplify.

1/Cos θ = Sec θ

Now we see that both sides of the equation are equal due to the reciprocal identity.

Radian and Degree Measure

Trigonometry

Trigonometry is the relationship of a triangles sides and angles.

(Trigon) - Triangle

(ometry) - measure

Trigonometry - The measurement of triangles

Trigonometry is used to measure triangles, but more specifically, right triangles.


Angles

An angle is defined as two rays with a common endpoint.

In the diagram above, an angle is formed by the rays P and R with endpoint Q.

With an angle, what is being measured is the amount of rotation between the two rays.

Angles can be measured in different units such as degrees, radians, grads, and revolutions.

Just to clear up a common misconception, ANGLES CAN BE NEGATIVE, they are vectors and their sign dictates direction of ray rotation.

Now that we know that angles are vectors, we need to define a start and end of the angle. The start of an angle will be one ray, called the Initial Side, and the end will be called the Terminal Side.

In this case theta is the name of the angle.

What if the angle was not labeled, how would you know which side was the terminal side?

This is why there is something called standard position, and it dictates the initial and terminal sides.

Standard Position - The initial side is on the x-axis and the vertex is at the origin.

There are four key words that are needed to know about a set of angles.

Congruent - The angles have the same measure.

Complementary - the sum of the angles is 90 degrees.

Supplementary - the sum of angles is 180 degrees.

Co-Terminal - The angles have the same terminal side. Ex. 40, 400, -320 degrees.

Here is an example of Co-Terminal Angles

All of the above angles have the same terminal side, which makes them Co-Terminal.

Arc Length

Arc length on a circle is usually denoted as "S" on a circle.

In the diagram above, an arc is formed by angle theta.

The formula to find the arc length when using degrees is as follows:

The formula to find arc length when using radians and ONLY RADIANS is as follows:

Conversions

Often in trigonometry, it will be necessary to convert Radians to Degrees and so on. The best way to do that is by using a conversion factor.

If I need 30 degrees in Radians, I will use a conversion factor to find it.