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Or maybe we have a distance and angle and need to "plot the dot" along and up: Questions like these are common in engineering, computer animation and more. The main functions in trigonometry are Sine, Cosine and Tangent They are simply one side of a right-angled triangle divided by another. It is the ratio of the side lengths, so the Opposite is about 0.7071 times as long as the Hypotenuse. Also try 120°, 135°, 180°, 240°, 270° etc, and notice that positions can be positive or negative by the rules of Cartesian coordinates, so the sine, cosine and tangent change between positive and negative also. Because the radius is 1, we can directly measure sine, cosine and tangent.For any angle "θ": (Sine, Cosine and Tangent are often abbreviated to Get a calculator, type in "45", then the "sin" key: sin(45°) = 0.7071... Here we see the sine function being made by the unit circle: Note: you can see the nice graphs made by sine, cosine and tangent. Here are some examples: Because the angle is rotating around and around the circle the Sine, Cosine and Tangent functions repeat once every full rotation (see Amplitude, Period, Phase Shift and Frequency).Trigonometry helps us find angles and distances, and is used a lot in science, engineering, video games, and more!
When we want to calculate the function for an angle larger than a full rotation of 360° (2 We can also find missing side lengths.
The general rule is: When we know any 3 of the sides or angles we can find the other 3 (except for the three angles case) See Solving Triangles for more details.
In other words, trigonometric equations may have an infinite number of solutions.
Additionally, like rational equations, the domain of the function must be considered before we assume that any solution is valid.
Trigonometric equations are, as the name implies, equations that involve trigonometric functions.
Similar in many ways to solving polynomial equations or rational equations, only specific values of the variable will be solutions, if there are solutions at all.In other words, we will write the reciprocal function, and solve for the angles using the function.Also, an equation involving the tangent function is slightly different from one containing a sine or cosine function.However, with trigonometric equations, we also have the advantage of using the identities we developed in the previous sections.When we are given equations that involve only one of the six trigonometric functions, their solutions involve using algebraic techniques and the unit circle (see [link]).Often, the angle of elevation and the angle of depression are found using similar triangles.In earlier sections of this chapter, we looked at trigonometric identities.The period of both the sine function and the cosine function is There are similar rules for indicating all possible solutions for the other trigonometric functions.Solving trigonometric equations requires the same techniques as solving algebraic equations.Often we will solve a trigonometric equation over a specified interval.However, just as often, we will be asked to find all possible solutions, and as trigonometric functions are periodic, solutions are repeated within each period.