Tangent Function. The tangent of an angle is the ratio of the opposite side and adjacent side. Tangent is usually abbreviated as tan. Tangent θ can be written as tan θ. Example: Calculate the value of tan θ in the following triangle. Solution: A review of the sine, cosine and tangent functions. Graphs of `y = a sin x` and `y = a cos x` 2. Graphs of `y = a sin bx` and `y = a cos bx` 2a. Frequency of Music Notes; 2b. Frequencies of Notes on a Piano: Learning object; 3. Graphs of `y = a sin(bx+c)` and `y = a cos(bx+c)` 4. Graphs of tan, cot, sec and csc; 5. Applications of Trigonometric Graphs; 6. Composite Trigonometric Graphs; 6a. Biorhythm Graphs; 7.
- Graphing Sine Cosine Tangent Comparison
- Graphing Sine And Cosine Examples
- Graphing Sine And Cosine Pdf
A sine wave made by a circle:
A sine wave produced naturally by a bouncing spring:
Graphing Sine Cosine Tangent Comparison
Plot of Sine
The Sine Function has this beautiful up-down curve (which repeats every 2π radians, or 360°).
It starts at 0, heads up to 1 by π/2 radians (90°) and then heads down to −1.
Plot of Cosine
Cosine is just like Sine, but it starts at 1 and heads down until π radians (180°) and then heads up again.
Plot of Sine and Cosine
In fact Sine and Cosine are like good friends: they follow each other, exactly π/2 radians (90°) apart.
Plot of the Tangent Function
The Tangent function has a completely different shape ... it goes between negative and positive Infinity, crossing through 0, and at every π radians (180°), as shown on this plot.
At π/2 radians (90°), and at −π/2 (−90°), 3π/2 (270°), etc, the function is officially undefined, because it could be positive Infinity or negative Infinity.
Inverse Sine, Cosine and Tangent
The Inverse Sine, Cosine and Tangent graphs are:
Inverse Sine
Inverse Cosine
Inverse Tangent
Mirror Images
Here is Cosine and Inverse Cosine plotted on the same graph:
Graphing Sine And Cosine Examples
Cosine and Inverse Cosine
They are mirror images (about the diagonal)!
The same is true for Sine and Inverse Sine and for Tangent and Inverse Tangent. Can you see this in the graphs above?
The relationship between the cosine and sine graphs is that the cosine is the same as the sine — only it’s shifted to the left by 90 degrees, or π/2. The trigonometry equation that represents this relationship is
Look at the graphs of the sine and cosine functions on the same coordinate axes, as shown in the following figure. The graph of the cosine is the darker curve; note how it’s shifted to the left of the sine curve.
y = sin
x and y= cos x on the same axes.”/>The graphs of the sine and cosine functions illustrate a property that exists for several pairings of the different trig functions. The property represented here is based on the right triangle and the two acute or complementary angles in a right triangle. The identities that arise from the triangle are called the cofunctionidentities.
The cofunction identities are as follows:
These identities show how the function values of the complementary angles in a right triangle are related. For example, cosθ = sin (90° – θ) means that if θ is equal to 25 degrees, then cos 25° = sin (90° – 25°) = sin 65°. This equation is a roundabout way of explaining why the graphs of sine and cosine are different by just a slide. You probably noticed that these cofunction identities all use the difference of angles, but the slide of the sine function to the left was a sum. The shifted sine graph and the cosine graph are really equivalent — they become graphs of the same set of points. Here’s how to prove this statement.
You want to show that the sine function, slid 90 degrees to the left, is equal to the cosine function:
Graphing Sine And Cosine Pdf
Replace cos x with its cofunction identity.
Apply the two identities for the sine of the sum and difference of two angles.
The two identities are
Substituting in the x’s and angles,
Simplify the terms by using the values of the functions.
So you see, the shifted sine graph is equal to the cosine graph.