Trig Functions

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Use Trigonometry to Measure the View of Satellite Cameras

Consider a satellite that orbits earth at an altitude of 750 miles. Earth has a radius of 3,950 miles. How far in any direction can the satellite's cameras see? The figure shows the satellite and the length [more…]

Sine and Cosine with Algebra

You can approximate, fairly accurately, the sine and cosine of angles with an infinite series, which is the sum of the terms of some sequence, or list, of numbers. Take note, however, that the series for [more…]

Domains and Ranges of Trigonometry Functions

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

The Sine Function: Opposite over Hypotenuse

When you're using right triangles to define trig functions, the trig function sine, abbreviated sin, has input values that are angle measures and output values that you obtain from the ratio opposite/hypotenuse [more…]

The Cosine Function: Adjacent over Hypotenuse

The trig function cosine, abbreviated cos, works by forming this ratio: adjacent/hypotenuse. In the figure, you see that the cosines of the two angles are as follows: [more…]

How to Locate Reference Angles

Each of the angles in a unit circle has a reference angle, which is always a positive acute angle (except the angles that are already positive and acute). By identifying the reference angle, you can determine [more…]

Assign Negative and Positive Trig Functions by Quadrant

The sine values for 30, 150, 210, and 330 degrees are 1/2, 1/2, –1/2, and –1/2, respectively. All these multiples of 30 degrees have a sin whose absolute value of 1/2 . The following rule and figure help [more…]

Use 6 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 Cosecant and Secant Functions

The cosecant function, abbreviated csc, is the reciprocal of the sine function and thus uses this ratio: hypotenuse/opposite. The hypotenuse of a right triangle is always the longest side, so the numerator [more…]

The Cotangent Function

The last reciprocal function is the cotangent,abbreviated cot. This function is the reciprocal of the tangent (hence, the co-). The ratio of the sides for the cotangent is adjacent/opposite. [more…]

Positive and Negative Angles on a Unit Circle

The unit circle is a platform for describing all the possible angle measures from 0 to 360 degrees, all the negatives of those angles, plus all the multiples of the positive and negative angles from negative [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…]

The Tangent Function: Opposite over Adjacent

The third 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: opposite/adjacent [more…]

How to Create a Table of Trigonometry Functions

The angles used most often in trig have trig functions with convenient exact values. Other angles don’t cooperate anywhere near as nicely as these popular ones do. A quick, easy way to memorize the exact [more…]

How to Find a Missing Coordinate on a Unit Circle

If you have the value of one of a point’s coordinates on the unit circle and need to find the other, you can substitute the known value into the unit-circle equation and solve for the missing value. [more…]

Find the Trigonometry Function of an Angle in a Unit Circle

You can determine the trig functions for any angles found on the unit circle — any that are graphed in standard position (meaning the vertex of the angle is at the origin, and the initial side lies along [more…]

Domain and Range of Tangent and Cotangent Trigonometry Functions

The tangent and cotangent are related not only by the fact that they’re reciprocals, but also by the behavior of their ranges. In reference to the coordinate plane, tangent is [more…]

Domain and Range of Cosecant and Secant Trigonometry Functions

The cosecant and secant functions are closely tied to sine and cosine, because they're the respective reciprocals. In reference to the coordinate plane, cosecant is [more…]

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