The word “trigonometry” might strike a note of dread with many people as they recall struggling through high school math classes, but the truth is that trigonometry comes in handy in a wide range of science and math applications.

Having a calculator for advanced math and trigonometry work is great, but it can’t help you out unless you understand what you’re doing and know how to use it. We’re going to talk a little about how to use a tangent calculator with degrees and minutes, and some of the principles and theory behind these higher-math functions.

**What Is a Tangent? **

Think for a moment about a right-angled triangle. The tangent can be calculated using the lengths of the sides of this triangle – the adjacent side is horizontal next to the angle you’re wanting to analyze, and then the opposite side is vertical, opposite to the angle you’re analyzing. The hypotenuse of this triangle, then, is involved with the definitions of cos and sin, but not that of tan. So let’s think of it this way:

**Tan = opposite/adjacent**

In other words, opposite and adjacent correspond to the lengths of the sides that are given those names. If the hypotenuse is a slope, then, the tan of the angle of that slope also describes the rise of that slope (aka vertical change) divided by the length of that slope (aka horizontal change). With this in mind, the tan of the angle can also be described as:

**Tan = sin/cos**

When applied to the specific angle that you’re interested in, the tangent of the angle will tell you what the tan function returns. This is a function also known as “arctan” or tan-1 – it reverses the tan function, returning the original angle when it’s applied to the tan of the angle. For the sin and cos functions, arcsin and arccos serve the same purpose.

Tangents can be converted to degrees by applying the arctan function to the tan of the angle you’re analyzing. This equation describes how to convert tangents to degrees:

**Angle in degrees = arctan**

In other words, the arctan function can be thought of as the reverse of the tan function.

## Using A Calculator To Analyze Sine, Tangent and Cosine as Angles

With a calculator (let’s go with the Texas Instruments TI-84 Plus for the sake of this discussion), it’s not hard to convert basic trigonometric functions and express them as angles, measured in degrees or radians. The TI-84 Plus in particular is capable of both directions, from angle to trigonometric measure and then back again. Here, we’ll use degrees instead of radians for consistency’s sake, but it’s no different from coming up with the same measurements in radians. All you need to do would be to set the calculator to the radians mode rather than degrees, as a first step.

## Making the Conversion from Trigonometric Functions to Degrees

Use the MODE key to set the calculator to Degrees mode, then press the Down arrow until you see the row with options “Degree” or “Radian”

Highlight “Degree” using the right arrow key, then press ENTER

Enter the inverse trigonometric function of the value you wish to see expressed as degrees

Press the 2ND key, then press they key for the trigonometric function in question

Example: to convert the sine of .5 into degrees, press 2ND followed by SIN. The display will then show sin^-1, or inverse sine. Follow that by entering .5 and closing parenthesis

Press ENTER and see the answer, expressed in degrees. Example: if sin^1 of .5 is entered, the calculator will give the answer of 30, which will be 30 degrees

Don’t forget the closing parenthesis

## Using a Graphing Calculator To Determine Cotangent

Remember that in trigonometry, the cotangent can be described as the reciprocal of the tangent. The formula for determining tangent involves dividing the opposite side by the adjacent side of the triangle. If the cotangent is the reciprocal, then, the formula that determines cotangent would involve dividing the adjacent side by the opposite side of the triangle. To put cotangent into a graphing calculator, the first step would be to know the angle (expressed in degrees) for which you’re trying to determine the cotangent.

Type 1 into the graphing calculator, followed by the division sign. The calculator will now be ready to perform a reciprocal calculation

Press the TAN button

Enter the angle for which you are determining the cotangent

## Calculating Arcsine on a Scientific Calculator

Remember that arcsine can be expressed as the inverse of the sine function.

For example, if you were calculating the arcsine of 3, you would press 2ND followed by SIN, and sin^-1 would appear. Press 3 and the equation would display as sin^-1(3). From there, just press ENTER to calculate the answer.

## Solving 3-Variable Linear Equations With a Calculator

Can you work your way through a system of linear equations by hand? Yes, but it’s a process that’s tedious, mistake-prone and time-consuming. Instead, let’s talk about how to solve linear equations or matrix equations with a calculator, referring back to our trusty TI-84 Plus graphing calculator.

This can be thought of as matrix A, multiplied by a vector of the unknowns and resolved with a vector B of constants. The calculator then inverts matrix A and multiplies A inverse and B to yield the unknowns in the equation.

A tangent line will only touch a curve at one point. A tangent line can be determined by using the point-slope method, aka slope-intercept. To put the slope-intercept equation in algebraic form, it would read y = mx + b, with “m” as the slope of the line and “b as the y-intercept, or the point where the tangent line meets the y-axis. To express this in algebraic form, this equation would be y -a0 = m(x – a1), with the slope of the line referred to as “m” and (a0, a1) as a point on the line.

**Wrapping Up**

If you’ve got a scientific calculator, it’s going to be invaluable for trigonometry and other higher math tasks. Once you’re familiar with how to use the calculator’s secondary functions such as SIN-1 and less-common functions (such as square root) and learn how to easily switch between degrees and radians, you’ll find these tasks are a lot easier and your calculator will serve as a true tangent calculator. Just remember to always include the close-parenthesis mark – leaving the right parenthesis off will likely return results that are far different from what you’ve been working toward.

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