Solving Trigonometric Equations and Identities

It is said that spies and other nefarious characters will carry many passports, enabling them to claim a different identity instantly. Despite the many identities, we know that all these passports are aliases of the same person.

Trigonometric identities are different ways of representing the same expression. They are used to solve a trigonometric equation when applied to the given scenario.

When a criminal is on the run, they will choose an Italian passport to assume a covert Roman identity for travel purposes. An identity is selected and applied to the expression for a solution to a trigonometric equation.

Let’s examine the fundamental identities before verifying them to solve trigonometric equations.

Trigonometric equations and their verified fundamental identities

Identities are the enablers that simplify complicate trigonometric expressions or equations. They are vital tools in the solution of trigonometric equations. Trigonometric identities work alongside factoring, finding special formulas, and using common denominators.

Like an algebraic equation, trigonometric equations are composed of basic formulas and properties of algebra. Perfect square and difference of square simplify working with expressions and trigonometric equations. It’s common knowledge that all trigonometric functions are closely related. That’s because they are all definitions of the unit circle, and their identities can be written in several ways.

Solving trigonometric identities

To solve trigonometric identities requires that you start with the more complicated part of the equation. You'll have to essentially rewrite the trigonometric expression until it’s transformed to become similar to the other part of the equation.

To obtain the desired results, you may have to expand or factor expressions while finding common denominators. You can also use any other algebraic strategy to transform the expression.

Exploring algebraic techniques for solving complicated trigonometric equations

Consider this function: f(x) = 2x² + x.

Solve f(x) = 0.

You already know that solving the function requires simple algebra. This will work out like;

2x² + x = 0

x (2x + 1) = 0

x = 0 or

x = − ½

Given the same scenario with g (t) = sin (t) and being asked to solve g (t) = 0. You can find the solution for this function using unit circle values.

These may include sin (t) = 0 for t = 0, π, 2π and others.

Using similar concepts, you can now consider the following functions and their composition.

f (g(t)) = 2(sin(t))² + (sin(t)) = 2sin²(t) + sin(t)

The equation created is called a polynomial-trigonometric function. To solve trigonometric equations using the like functions, use identities, the quadratic formula, and algebraic techniques such as factoring.

Basic quadratic formulas

Six or more ratios exist that can derive trigonometric elements. They are known as trigonometric functions. These are sine, cosine, secant, co-secant, tangent, and co-tangent.

Identities and functions of trigonometry are derived using the right-angled triangle as a reference. They appear as;

• Sin θ: Equals the Opposite Side divided by the Hypotenuse
• Cos θ: Equals the Adjacent Side divided by the Hypotenuse
• Tan θ: Equals the Opposite Side divided by the Adjacent Side
• Sec θ: Equals the Hypotenuse divided by the Adjacent Side
• Cosec θ: Equals the Hypotenuse divided by the Opposite Side
• Cot θ: Equals the Adjacent Side divided by the Opposite Side

Reciprocal identities

In some cases, a single trigonometric equation will use a variety of reciprocal identities. These are given as;

• sin θ = 1 divided by cosec θ
• cot θ = 1 divided by tan θ
• tan θ = 1 divided by cot θ
• cosec θ = 1 divided by sin θ
• cos θ = 1 divided by sec θ
• sec θ = 1 divided by cos θ

Reciprocal identities arise from a right-angled triangle. When the base and height are given, you can find the sin, cos, tan, sec, cos, and cot values with trigonometric formulas.

By using trigonometric functions, reciprocal trigonometric identities can also be derived.

Periodicity identities

Also called co-function identities, the periodicity formulas are used for shifting angles by π/2, π, and 2π and so forth.

Periodicity formulas are useful in calculating complex geometry, evaluating trigonometric functions, and proving other identities. The first two are; sin (90°−x) = cos x, and cos (90°−x) =sin x, and are the most commonly used.

Since trigonometric identities are cyclic, they repeat themselves following the periodicity constant. For different trigonometric identities, the radian constant for periodicity is different.

Tan 45° = tan 225°, and the same is applicable for cos 45° = cos 225°.

Co-function identities

You can also represent the co-function of periodic identities in degrees. These include sin (90°−x) = cos x, cos (90°−x) = sin x, tan (90°−x) = cot x, cot (90°−x) = tan x and so on.

Co-function or periodic identities relate to trigonometric function co-pairs at x and π/2. For instance, sin (π/2 - x) = cos(x), cos (π/2 - x) = sin(x), tan (π/2 - x) = cot(x), and cot (π/2 - x) = tan(x).

Double angled identities

The angle addition formula can derive double angled identities. They are not principally hard to memorize and usually come in handy in a lot of trigonometric equations.

You can use the double angled identities to find the cosine and sine of 2x in terms of the cosines and sines of x following from the angle-sum formula.

Angle sum identities

The sine area formula for a triangle is A = ½ ⋅ ab sin C. Angle sum identities tell how to find the sine and cosine of x + y when the cosines and sines of x and y are given.

Half angled identities

These may look intimidating, but they are simple to derive. The double angled formula can further generate half angled identities.

You can use the half angled identities to find the sine and cosine of x/2. This is in terms of the cosines or sines given for x following after the double angled formulas.

Negative angle identities

The negative-angle identities are based on the unit circle and are often called odd, even identities. You can find the trigonometric functions at –x when the identities of x relate to values at opposing angles –x and x.

For instance;

sin(−t) = −sin(t)cos(−t) = cos(t)tan(−t) = −tan(t)csc(−t) = −csc(t)sec(−t) = sec(t)cot(−t) = −cot(t)

Alongside reciprocal identities, you can use these to solve a single equation.

Example

You are given a trigonometric equation that looks a lot like a quadratic equation.

2sin2 (t) + sin (t) = 0

The problem requires all solutions with 0≤t<2π. It is also called a quadratic in-sine equation because of the sin (t) instead of a quadratic variable.

Use the quadratic formula or factoring techniques as with all quadratic equations. By factoring out the common sin (t) factor, the expression factors well.

sin (t)(2sin(t)+1) = 0

If either factor is zero, you know that the product on the left equals zero. It is also called the zero product theorem, and it enables you to break the equation into two expressions.

sin (t) = 0 or,

2sin (t) + 1 = 0

These equations can then solved independently.

sin (t)= 0t = 0 or,

t = π

2sin(t)+1 = 0sin(t) = −12t=7π6 or,

11π6

It will give you four solutions for the 0≤t<2π: t=0, π, 7π6, 11π6 equation. To check if the answers are reasonable, you can compare the zeros after graphing the function.

Conclusion

To solve trigonometric equations, you need to use identities and reference angles to memorize alongside algebra.

These equations require that you think and have a good grasp of the​ first quadrant trig-ratio value and the workings of the unit circle. Be ready to identify the various trigonometric functions in the first period or the relationship between degrees and radians.

One of the key concepts to take with you is that multiple representations of a trigonometric expression exist. A verified identity will illustrate how to simplify the equation by rewriting the expression.

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