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### 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…]

### How to Use Heron's Formula to Find the Area of a Triangle

You can find the area of a triangle using Heron's Formula. Heron's Formula is especially helpful when you have access to the measures of the three sides of a triangle but can't draw a perpendicular height [more…]

### 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…]

### Reciprocal Identities

A big advantage of trig expressions and equations is that you can adjust them in so many ways to suit your needs. The basic reciprocal identities here are the ones people use most frequently. [more…]

### Sum-to-Product Identities

The sum to product identities are useful for modeling what happens with sound frequencies. Think of two different tones represented by sine curves. Add them together, and they beat against each other with [more…]

### Basic Trigonometric Figures

Segments, rays, and lines are some of the basic forms found in geometry, and they're almost as important in trigonometry. You use those segments, rays, and lines to form angles. [more…]

### Basic Trigonometric Angles

When two lines, segments, or rays touch or cross one another, they form an angle or angles. In the case of two intersecting lines, the result is four different angles. When two segments intersect, they [more…]

### Basic Trigonometric Triangles

All on their own, angles are certainly very exciting. But put them into a triangle, and you've got icing on the cake. Triangles are one of the most frequently studied geometric figures. The angles that [more…]

### Radius, Diameter, Circumference, and Area of Circles

A *circle* is a geometric figure that needs only two parts to identify it and classify it: its *center* (or middle) and its *radius* (the distance from the center to any point on the circle). After you've chosen [more…]

### Chords versus Tangents of Circles

You show the diameter and radius of a circle by drawing segments from a point on the circle either to or through the center of the circle. But two other straight figures have a place on a circle. One of [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…]

### How to Solve for a Missing Right Triangle Length

The Pythagorean theorem states that *a*^{2} + *b*^{2} = *c*^{2} in a right triangle where *c* is the longest side. You can use this equation to figure out the length of one side if you have the lengths of the other two [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…]

### Product-to-Sum Identities

The trig product-to-sum identities look very much alike. You have to pay close attention to the subtle differences so that you can apply them correctly. Even though the product looks nice and compact, [more…]