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### Commonly Used Values of Selected Trig Functions

When performing transformations in trig functions, such as rotations, you need to use the numerical values of these functions. Here are some of the more commonly used angles. [more…]

### How to Calculate Trigonometry Functions Using Any Circle

When determining the trig function values of angles graphed in standard position on a circle whose center is at the origin, you don’t have to have a unit circle to calculate coordinates. You can use a [more…]

### A Quick Table for the Three Reciprocal Trig Functions

Reciprocal functions have values that are reciprocals, or flips, of the values for their respective functions. The reciprocal of sine is cosecant, so each function value is the reciprocal of the corresponding [more…]

### The Secant Function

The *secant* function, abbreviated *sec*, is called a reciprocal function because it uses the reciprocal of the trig function, cosine. The secant function uses the ratio [more…]

### Assign Negative and Positive Trig Function Values by Quadrant

The first step to finding the trig function value of one of the angles that’s a multiple of 30 or 45 degrees is to find the reference angle in the unit circle. When the reference angle comes out to be [more…]

### Domain and Range of Cosecant and Secant Trigonometry Functions

The *domain* of a cosecant or secant trig function consists of all the input values that the function can handle — the way the function is defined. Of course, you want to get output values [more…]

### Use One Function to Solve for Another Function

Sometimes you have to solve for a trig function in terms of another function. In the following example, the cosine of angle lambda is 12/13. What are the values of the sine and tangent of lambda? [more…]

### How to Remember the Ratios for the Three Basic Trig Functions

The legend of SohCahToa is an easy way to remember the ratios for the three basic trig functions. Sure, the story is pretty lame, but you'll find it very useful when trying to remember the ratios for the [more…]

### Labeling the Sides of a Right Triangle

A right triangle has two shorter sides, or legs, and the longest side, opposite the right angle, which is always called the *hypotenuse.* The two shorter sides have some other special names, too, based on [more…]

### A Quick Table for the Three Reciprocal Trigonometry Functions

You know that the reciprocal functions have values that are *reciprocals,* or flips, of the values for their respective functions. The reciprocal of sine is cosecant, so each function value for cosecant [more…]

### How to Place Points on a Unit Circle

The *unit circle* is a circle with its center at the origin of the coordinate plane and with a radius of 1 unit. Any circle with its center at the origin has the equation [more…]

### How to Compute Reference Angles in Degrees

Solving for the reference angle in degrees is much easier than trying to determine a trig function for the original angle. To compute the measure (in degrees) of the reference angle for any given angle [more…]

### How to Compute the Reference Angles in Radians

Solving for the reference angle in radians is much easier than trying to determine a trig function for the original angle. To compute the measure (in radians) of the reference angle for any given angle [more…]

### How to Calculate Trigonometry Functions of Angles Using the Unit Circle

Calculating trig functions of angles within a unit circle is easy as pie. The figure shows a unit circle, which has the equation *x*^{2} + *y*^{2} = 1, along with some points on the circle and their coordinates. [more…]

### How to Calculate Coordinates at the Origin on Any Unit Circle

You don’t need a unit circle to use this coordinate business when determining the function values of angles graphed in standard position on a circle. You can use a circle with any radius, as long as the [more…]

### Domain and Range of Sine and Cosine Functions

The sine and cosine functions are unique in the world of trig functions, because their ratios always have a value. No matter what angle you input, you get a resulting output. The value you get may be 0 [more…]

### Angle of Elevation and Angle of Depression in Trigonometry Functions

Mathematical problems that require the use of trig functions often have one of two related angles: the angle of elevation or the angle of depression. The scenarios that use these angles usually involve [more…]

### Using Trigonometry to See if a Ladder Reaches a Window

Every day, people use trigonometry to measure things that they can't reach. How high is that building? Will this ladder reach to the top of that tree? By using the appropriate trig functions, you can find [more…]

### How to Determine the Height of a Tree

Which trig function should you use to determine the height of a tree? Suppose you're flying a kite, and it gets caught at the top of a tree. You've let out all 100 feet of string for the kite, and the [more…]

### How to Measure the Distance between Two Rooftops

You can use trig functions to measure the distance between the rooftops on buildings. Why would you need to do this? Well, Jumping Jehoshaphat makes his living by jumping, on his motorcycle, from building [more…]

### How to Measure the Slope of a Road

Land surveyors use trigonometry and their fancy equipment to measure things like the slope of a piece of land (how far it drops over a certain distance). Have you ever noticed a worker along the road, [more…]

### How to Determine the Altitude of a Balloon

You can use trigonometry functions to determine the altitude of a balloon. Cindy and Mindy, standing a mile apart, spot a hot-air balloon directly above a particular point on the ground somewhere between [more…]

### How to Determine the Vertical Distance Travelled by a Rocket

Trig functions come in handy if you work for NASA or need to measure the vertical distance travelled by a rocket. In this example, a rocket is shot off and travels vertically as a scientist, who's a mile [more…]

### How to Use Trigonometry to Find the Area of a Triangle

Sometimes, finding a measure isn't so easy. You may have to deal with an irregular shape, like a triangle, or even calculate your way around a fixed object. Whatever the case, you can use trigonometry [more…]

### Determining if a Long Object Will Fit around a Hallway Corner

Here's an application of trigonometry that you may very well be able to relate to: Have you ever tried to get a large piece of furniture around a corner in a house? You twist and turn and put it up on [more…]