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More details on trigonometric equations

There’s no doubt that trigonometry plays a significant role in mathematics and in other related fields, including technology. The word "trigonometry" consists of two Greek words - "trigonon" and "metron", standing for respectively "triangle" and "measure". Trigonometry can be defined as a study of the relationship between the ratios of angles and sides in context with right angled triangles.

In general, trigonometry deals with six basic trigonometric functions. These are secant (sec), sine (sine), tangent (tan), cosine (cos), cosecant (csc or cosec) and cotangent (cot). Cot, sec and cosec appear to be the inverse functions of tan, cos and sin. Keep reading to learn more, including trigonometric equations.

Real-life applications of trigonometry

Perhaps, you wonder, why on earth you need to study these naughty trigonometric equations or any other related stuff. The matter is that trigonometry is widely used in everyday life and we can hardly enumerate all its areas of application. For instance, in navigation trigonometry is actively utilized to determine distances and locations, using and trigonometric equations, of course. Furthermore, you can come across trigonometric equations in such fields as electrical engineering, mechanical engineering, physics, music, astronomy, economy and biology. Here below we’ll get familiar with persuasive and vivid examples of the use of trigonometry. You’ll see how trigonometric equations work.

Horizontal ray

It’s a ray parallel to the surface of the earth. It emerges from the eye of the observer. That’s why it’s called horizontal.

Ray of vision

That’s a ray running from the observer’s eye towards the object under observation. Its alternative name is ray of sight.

Angle of elevation

If a particular object under observation appears to be above the horizontal ray, coming through the point of observation, then the measure of the angle generated by the ray of vision and the horizontal ray is angle of elevation.

Angle of depression

If the object being observed appears to be below the horizontal ray coming through the point of observation, the measure of the angle generated by the ray of vision and the horizontal ray is dubbed angle of depression.

Applications of right triangle trigonometry

Do you want to see more real-life applications of trigonometry and trigonometric equations? We’ll do it right now! As we’ve already told above trigonometry deals with triangles and their relationships. The given math discipline is definitely good at calculating the height of any object, including mountains towers, buildings. Finding the distance of the shore from a certain point in the sea is also another duty of trigonometry. It gives perspective on real world events. For instance, it’s actively employed in satellite systems, architecture, engineering, astronomy, geography as well as other fields.

Applications of trigonometry in engineering

As we’ve already mentioned above, trigonometry is all about the angles and sides of triangles. The matter is that engineering problems are stuffed with such relationships. Engineers, working in various fields of industry are used to employing the fundamentals of trigonometry, including trigonometric equations when it comes to designing bridges, building structures and solving scientific problems. Trigonometry and in particular trigonometric equations are extremely important for those engineers, who are used to work with magnetic and electrical fields and waves.

Applications of trigonometry in astronomy

Certainly, astronomers can’t do without trigonometric equations too. They use them when it comes to calculating the distance to the stars. Furthermore, trigonometric equations help to measure distance between universe objects located at great distances.

Applications of trigonometry in architecture

In architecture, trigonometry plays a crucial role. The given discipline is employed to design a building. To be exact, it actually predetermines geometrical patterns and how much labor and materials is going to be utilized to build this particular tower or anything else.

Types of trigonometric equations

As for trigonometric equations, we can define them as the equations, containing one or several trigonometric functions. There’re several types of trigonometric equations:

  • Quadratic equations
  • Linear equations
  • Higher order equations
  • Inverse equations

All the algebraic operations can be successfully applied when it comes to solving trigonometric equations.

By the way, such type of equations are mostly quadratic or linear by nature. As for linear equations, they’re traditionally solved for the variable by simply isolating this stuff. Factoring is actively utilized when it comes to solving quadratic equations. Solving the equation suggests finding all the replacements for the variable, making the equation true.

Classification of trigonometric equations

Trigonometric equations are often classified this way:

  • Equations of quadratic form
  • Equations of removed square roots
  • Equations dealing with more than two trigonometric functions
  • Equations dealing with the trigonometric functions of several angles
  • Equations of opposite functions

Trigonometric equations: must-have steps to solve

First of all, you require stating all the functions into a certain trigonometric function with the help of dissimilar trigonometric identity.

Secondly, you’re expected to move the constants as well as numbers to the right hand side of the equation as well as trigonometric function to left hand side of the equation.

Thirdly, you require explaining the trigonometric function and then getting the opposite trigonometric function on both sides of your equation in order to calculate the unknown angle.

Trigonometric equations examples

A great number of learners think that they’ve successfully solved a trig equation once one answer is available. Unfortunately, they often forget that in most cases just one answer isn’t enough for such equations. As a result, they lose marks.

As for the overall process of tackling general trigonometric equations, we can’t offer you a clear-cut solution. Don’t even try to look for magic rules leading to the right solution on a regular basis. The process normally suggests the use of algebraic manipulation, identities, and certainly trial and error.

If your equation comes with more than one function, you should employ identities as well as algebraic manipulation (for instance, factoring) in order to rewrite the overall equation in terms of just one trigonometric function.

You should look for expressions in the quadratic form and have them solved by factoring. Keep in mind that not all equations boasts solutions, however, those, which do, are often solved by means of appropriate identities as well as algebraic manipulation. Seek suitable patterns. Nothing could be a proper substitute to your own experience.

Don’t forget to find other solutions. We’re talking about the other angles, satisfying your equation.

Your calculator needs to be set to degrees if you want to properly solve your equations.

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