Trigonometry

Trigonometry calculates the horizontal and vertical distance of a unit circle point that is specified by an angle.  The angle is the perimeter distance to the point on the circle, from a predetermined starting point on the circle, which in our examples will be the East (3 O’Clock) position.

As explained in the previous lesson o, we will be using a right handed coordinate system with angles measured counter-clockwise (CCW) from East:

We divide the coordinate system into four quadrants, in the counter-clockwise order of traversing the unit circle perimeter.  The first quadrant includes North East directions (NE), the second quadrant NW, the third quadrant SW, and the fourth quadrant SE:

The first quadrant is subtended by the angle 𝛑╱𝟐 (90 degrees):

We assign the symbol theta ( 𝜃 ) to denote an angle that is the perimeter distance along the unit circle measured counter-clockwise (CCW) from East:

If the angle is in the first quadrant, as shown here, it is called an acute angle o.  Trigonometry finds the horizontal and vertical position of any point on the unit circle at any angle (in any quadrant).

In this discussion, we denote the horizontal (eastward) position of a point as x, and the vertical (northward) position as y:

The horizontal distance x is referred to as the cosine (“cos”) of the point at theta, and the vertical distance y is referred to as the sine (“sin”) of the point:

x  =  cos𝜃
y  =  sin𝜃

Cosine and sine are pronounced co-sign and sign respectively (not spelled that way, just pronounced that way):

Pronounciation of “sine”. [Wikimedia]

The following formulas may be used to calculate cosine and sine:

cos𝜃 = 1 − (𝜃2 ⁄ 2!) + (𝜃4 ⁄ 4!) − (𝜃6 ⁄ 6!) + ⋯
sin𝜃 = 𝜃 − (𝜃3 ⁄ 3!) + (𝜃5 ⁄ 5!) − (𝜃7 ⁄ 7!) + ⋯

where the exclamation mark (!) is the factorial symbol, which multiplies a number with all positive integers that are less than the number. For example:

5! = 5 × 4 × 3 × 2 × 1

These types of formulas are best left to calculators and computers. Modern calculators have buttons for sine and cosine.

The inverse of the cosine and sine functions are arccosine and arcsine, which give us the angle of a given horizontal and vertical distance respectively.

The inverse function arccos may also be denoted as cos−1, and arcsin may be denoted sin−1

arccos  ↔  cos−1

arcsin  ↔  sin−1

Formulas are available for calculating inverse trigonometric functions.

But, again, such tedious numerics are best left to calculators and computers. We usually only need to know the sine or cosine, or arc length (angle) of a sine or cosine, not how it’s calculated.

On calculators, often the same buttons are used for forward and inverse trigonometric functions, except that a shift key is pressed (and released) beforehand to notify the calculator to use inverse instead of forward trigonometry.

For example, on the Texas Instruments handheld calculator pictured, punching in a number and then pressing the SIN key calculates the sine of that number, but pressing (and releasing) the 2ND key right before pressing the SIN key invokes the arcsine function:

On this calculator, a blue description (upper left above each button) is invoked if the 2ND key is pressed beforehand. The green description (a text character to the right of the blue description) is invoked if the ALPHA key is pressed beforehand (to enter a letter of the alphabet).

For a calculator application program on a computer, if the buttons are on-screen (not actual physical buttons), the description on a button may change:

Exercise

Problem: Calculate the sine of 60 degrees using the Microsoft Windows calculator.

Solution: The Microsoft Wndows calculator varies depending on which version of the operating system you are using. This example will use a slightly different version than illustrated above (with some of the feature differences noted).

If the calculator does not have “sin” and “cos” buttons, click on View (or the menu icon in the other calculator) and select “Scientific”.

Then make sure the type of angles is set to Degrees, as shown here. In the other on-screen calculator illustrated earlier, if it says RAD or GRAD, click/tap on that icon until it says DEG.

Next, press the ‘6’ number key, then the ‘0’ number key on your keyboard (or click/tap on the ‘6’ and ‘0’ screen buttons on the calculator).

Then click or tap on the “sin” calculator screen button (not the “sinh” button which would be for hyperbolic functions).

To find the inverse, click on “Inv” then click on the arcsine button.