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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…]

### Use Six Different Ratios of a Right Triangle

Each of the three sides of a right triangle — hypotenuse, opposite, and adjacent — has a respective length or measure. And those three lengths or measures form six different ratios. Check out the following [more…]

### The Sine Function: Opposite over Hypotenuse

When you’re using right triangles (triangles with right angles in them) to define trig functions, the trig function *sine*, abbreviated *sin*, has input values that are angle measures and output values that [more…]

### How to Compute the Reference Angles in Radians

If you need to compute the measure (in radians) of the reference angle for any given angle θ, you can use the rules in the following table. [more…]

### Use Coordinates of Points to Find Values of Trigonometry Functions

One way to find the values of the trig functions for angles is to use the coordinates of points on a circle that has its center at the origin. Letting the positive [more…]

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

Calculating trig functions on angles using the unit circle is easy as pie. The following 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 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…]

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

Sometimes, it’s hard to find the area of a triangle, especially when it’s not convenient to measure its height. When this happens, you can use trig to modify this traditional formula for the area of a [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…]

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

Here’s an application of trigonometry that you may 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 end, but to [more…]

### How to Measure the Slope of a Road

Trigonometry functions have everyday applications. For example, you can use a trigonometry function to measure the slope of the surface you are standing on. [more…]

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

Mathematical problems that require the use of trig functions often refer to one of two special 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

You can use a trigonometry function to find the height of a tree while standing on the ground. For example, suppose you’re flying a kite, and it gets caught at the top of a tree. You’ve let out all 100 [more…]

### How to Measure the Distance between Two Rooftops

Trigonometry functions have everyday applications. You can use a trig function to measure the distance between two rooftops — you just have to be standing on one of them. [more…]

### How to Create a Table of Trigonometry Functions

In case your scientific calculator decides to conk out on you, you can construct a table that shows the exact values of trigonometry functions for the most commonly used angles. A bonus to this situation [more…]

### The Tangent Function: Opposite over Adjacent

The trig function, *tangent*, is abbreviated *tan*. This function uses just the measures of the two legs and doesn’t use the hypotenuse at all. The tangent is described with this ratio: [more…]

### Use One Function to Solve for Another Function

Sometimes you have to solve for a trigonometry function in terms of another function. For example, you may be given the cosine of an angle, and from that, you need to determine the sine and tangent of [more…]

### The Cosecant Function

The *cosecant* function, abbreviated *csc*, is called a reciprocal function because it is the reciprocal of one of the three basic trig functions, sine. The cosecant function uses the ratio [more…]

### The Cotangent Function

The *cotangent* function, abbreviated *cot*, is called a reciprocal function because it uses the reciprocal of the trigonometry function, tangent (hence, the [more…]

### How to Place Points and Fractions on a Unit Circle

To work with trigonometry functions, it’s helpful to be able to place points and fractions on a unit circle. The unit circle is a circle with its center at the origin of the coordinate plane and with a [more…]

### Negative and Positive Angles Cutting a Circle

The circle with its center at the origin is a platform for describing all the possible angle measures from 0 to 360 degrees, plus all the negatives of those angles, and plus all the multiples of the positive [more…]

### How to Locate Reference Angles

Each of the angles positioned with its center at the origin has a *reference* angle, which is always a positive acute angle. By identifying the reference angle, you can determine the function values for [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…]