Trignometric Functions	Domain	Interval (Range)
[Tex]\sin{\theta}[/Tex]	[Tex](−∞, ∞)[/Tex]	[Tex][-1,1][/Tex]
[Tex]\cos{\theta}[/Tex]	[Tex](−∞, ∞)[/Tex]	[Tex][-1,1][/Tex]
[Tex]\tan{\theta}[/Tex]	[Tex]\mathbf{R} -\ (\mathbf{2n}\ +\ \mathbf{1})\frac{\pi}{\mathbf{2}}[/Tex]	[Tex](−∞, ∞)[/Tex]
[Tex]\csc{\theta}[/Tex]	[Tex]\mathbf{R} -\ (\mathbf{n\pi})[/Tex]	[Tex](-∞, -1] \bigcup\ [1, ∞)[/Tex]
[Tex]\sec{\theta}[/Tex]	[Tex]\mathbf{R} -\ (\mathbf{2n}\ +\ \mathbf{1})\frac{\pi}{\mathbf{2}}[/Tex]	[Tex](-∞, -1] \bigcup\ [1, ∞)[/Tex]
[Tex]\cot{\theta}[/Tex]	[Tex]\mathbf{R} -\ (\mathbf{n\pi})[/Tex]	[Tex](−∞, ∞)[/Tex]

Read More,

• Domain and Range of Trigonometric Functions
• Trigonometry Table

Trignometric Identities

Trigonometric identities are mathematical statements regarding angles that remain valid regardless of their location in the unit circle. Simplifying formulas and solving trigonometric equations requires these identities to be applied as tools.

• Pythagorean Identities
• Angle Sum and Difference Identities
• Double Angle Identities
• Half Angle Identities
• Product-to-Sum Identities
• Sum-to-Product Identities
• Inverse Trigonometric Identities

Solving Trigonometric Equations

Trigonometric equations are equations involving trigonometric functions. Solving these equations often requires the use of trigonometric identities and inverse trigonometric functions. Trigonometric equations, like ordinary polynomial equations, have solutions that are referred to as principle solutions and general solutions.

Trigonometric equations are mathematical formulas that involve sine, cosine, and tangent functions and are set equal to a value. These equations frequently have variables for angles, usually represented by the symbols [Tex]\theta[/Tex] or x, which need to be solved to determine the angles to make the equation true.

Steps to Solve a Trigonometric Equation

Step 1: Transform to a Single Trigonometric Ratio: Convert the given trigonometric equation into an equation involving a single trigonometric function (sin, cos, tan).

Step 2: Simplify Angles: If the equation involves multiple angles or submultiple angles, simplify it to a single angle using trigonometric identities.

Step 3: Rewrite as a Standard Equation: Express the trigonometric equation in the form of a polynomial, quadratic, or linear equation.

Step 4: Solve the Equation: Solve the equation as you would any other, then determine the trigonometric ratio.

Step 5: Find the Solution: The solution to the trigonometric equation is given by the angles corresponding to the trigonometric ratio or the value of the trigonometric ratio itself.

Principle Solution of Trigonometric Equations

A trigonometric equation’s primary solution is the answer that falls inside a given range, usually between 0° and 360° or 0 and 2π radians.

When determining alternative solutions, this primary solution—that represents the equation’s primary or fundamental solution—is often used as a point of reference.

Applications of Analytic Trigonometry

Trigonometry’s analysis is not merely a hypothetical area but entails a lot of applications. On different domains including physics, engineering and even biology we can find these utilizations.

Few of the applications are:

• Wave Motion and Oscillations
• Electrical Circuits
• Architecture and Construction
• Navigation and Astronomy

Wave Motion and Oscillations

Sinusoidal waves: If we consider a sinusoidal signal y(t) having an amplitude A, angular frequency [Tex]\omega[/Tex], and phase of quantity then we can represent the signal as:

[Tex]y(t)=A\sin(\omega t+ \theta)[/Tex]

Electrical Circuits

Trigonometric equations are used for the analysis and resolution of issues concerning electric circuits. In addition to this, they are mainly applied in investigating alternating current (AC) circuits. An AC circuit is one containing voltages or current that changes sinusoidally over a specified period of time; hence any trigonometric function is a true representation to such movements.

An AC voltage source in general form is given as [Tex]V(t)=V_{0}\sin{(\omega t)}[/Tex], where [Tex]V_{0}[/Tex] is the peak voltage and ? is the angular frequency.

Architecture and Construction

• Measuring Height with a Sextant: A sextant is a tool that measures the angle between two visible objects. It can be used to determine the height of a building.

Navigation and Astronomy

• Celestial Navigation with a Sextant: A sextant is a crucial tool in celestial navigation, used to measure the angle between a celestial object (such as the sun, moon, or stars) and the horizon.
• Distance Between Two Points on Earth: To find the distance between two points on the Earth’s surface, we can use the Haversine formula, which involves trigonometric functions.
• Calculating the Distance to a Star: Astronomers use trigonometry to calculate the distance to stars using the parallax method.

Examples on Analytic Trigonometry

Example 1: [Tex]\mathbf{2\sin{\theta} + 1 = 0}[/Tex]

Solution:

[Tex]\mathbf{2\sin{\theta} + 1 = 0} \Rightarrow \mathbf{\sin{\theta} = -\frac{1}{2}}[/Tex]

[Tex]\Rightarrow \theta = \mathbf{\sin^{-1}} (-\frac{1}{2})[/Tex]

Using identities: [Tex]\mathbf{\sin{(-\theta)} = -\sin\theta}\: and\\ \mathbf{\sin{(-30\degree)}} = -\mathbf{\sin{(30\degree)}} = -\frac{1}{2}[/Tex]

[Tex]\Rightarrow \mathbf{\theta=sin^{-1}(\sin{(-30\degree)})}\\ \Rightarrow \mathbf{\theta=-30\degree}[/Tex]

Also, as we know,

[Tex]\mathbf{\sin\theta}=\mathbf{\sin{(\pi-\theta)}}=\mathbf{\sin{(180\degree-\theta)}}[/Tex]

Therefore,

[Tex]\mathbf{\theta}= -30\degree and\: 210\degree[/Tex]

Principal argument of trig functions lies in [Tex]0<\theta<2\pi[/Tex]:

[Tex]\mathbf{\theta(principle)=210\degree}[/Tex]

Eample 2: [Tex]\mathbf{\cos^2{\theta} – \frac{1}{2} = 0}[/Tex]

Solution:

[Tex]\mathbf{\cos^2{\theta} = \frac{1}{2}}[/Tex]

[Tex]\Rightarrow \mathbf{\cos{\theta} = \pm\frac{1}{\sqrt2}}[/Tex]

[Tex]\mathbf{\theta} = \mathbf{\frac{\pi}{4},\frac{3\pi}{4},\frac{5\pi}{4},…}[/Tex]

Example 3: [Tex]\mathbf{\cos{\theta}}=\mathbf{\tan{\theta}}[/Tex], solve for [Tex]\theta[/Tex].

Solution:

[Tex]\mathbf{\cos{\theta}}=\mathbf{\tan{\theta}}\\ \Rightarrow \cos{\theta}=\frac{\sin\theta}{\cos\theta}\\ \Rightarrow \cos^2{\theta}=\sin{\theta}[/Tex]

[Tex]\Rightarrow 1 – \sin^2{\theta}=\sin{\theta}, (\because \cos^2{\theta}+\sin^2{\theta}=0)[/Tex]

[Tex]\Rightarrow \sin^2{\theta}+\sin{\theta}-1=0[/Tex]

Let [Tex]\sin{\theta}=x[/Tex] then,

[Tex]x^2+x-1=0[/Tex]

Solving this equation gives,

[Tex]x=\frac{-1\pm\sqrt{5}\:}{2}=-1.618,0.618[/Tex]

Now, [Tex]-1.618<-1[/Tex], which is not possible to be [Tex]\sin{\theta}=x=-1.618[/Tex]. Thus we must have [Tex]\sin{\theta}=x=0.618[/Tex].

Hence, the value of [Tex]\theta[/Tex], following the steps as in solved example 1 we have:

[Tex]\theta=\sin^{-1}{(0.618)}=\sin^{-1}{(\pi-0.618)}[/Tex]

[Tex]\Rightarrow \theta=0.666 rad[/Tex]

Example 4: [Tex]\cos{3\theta}=1/2[/Tex], solve for [Tex]\theta[/Tex].

Solution:

[Tex]3\theta=\cos^{-1}{(\frac{1}{2})}=\frac{\pi}{3}[/Tex]

[Tex]\Rightarrow \theta=\frac{\pi}{3\times3}=\frac{\pi}{9}[/Tex]

Since, [Tex]cosine[/Tex] is an even function therefore [Tex]\cos{(-\theta)}=\cos(\theta)[/Tex], which gives

[Tex]\theta=\frac{\pi}{3}\:or\:-\frac{\pi}{3}[/Tex]

Example 4: [Tex]\sin{2\theta}=\sin{\theta}[/Tex], solve for [Tex]\theta[/Tex]

Solution:

\sin{2\theta}=\sin{\theta}

Using the double angle formula ([Tex]\sin{2\theta}=2\sin{\theta}\cos{\theta}[/Tex]), gives,

[Tex]\Rightarrow 2\sin{\theta}\cos{\theta}=\sin{\theta}[/Tex]

[Tex]\Rightarrow \sin{\theta}(2\cos{\theta}-1)=0[/Tex]

Therefore,

[Tex]\Rightarrow \sin{\theta}=0\:or\:\cos{\theta}=\frac{1}{2}[/Tex]

[Tex]\theta=0,\pi\:\:\:\:or\:\:\:\:\theta=\pm \frac{\pi}{3}[/Tex]

Practice Questions on Analytic Trigonometry

Q1. [Tex]\sin{\theta}=\tan{\theta}[/Tex], solve for [Tex]\theta[/Tex].

Q2. [Tex]2\sec{\theta}=1[/Tex], solve for [Tex]\theta[/Tex].

Q3. [Tex]2\cos{\theta}-3=-5[/Tex], solve for [Tex]\theta[/Tex].

Q4. [Tex]2\sin^{2}{\theta}-1=0, 0\leq\theta<2\pi[/Tex], solve for [Tex]\theta[/Tex].

Q5. [Tex]\tan{(\theta-\frac{\pi}{2})}=1, 0\leq\theta<2\pi[/Tex], solve for [Tex]\theta[/Tex].

Q6. [Tex]2(\tan{x}+3)=5+\tan{x},0\leq x<2\pi[/Tex] solve for [Tex]\theta[/Tex].

Q7. [Tex]2\sin^{2}{θ}−5\sin{θ}+3=0,0\leq\theta\leq2\pi[/Tex] solve for [Tex]\theta[/Tex].

Q8. Solve [Tex]\sin{2θ}=2\cos{\theta}+2, 0\leq\theta\leq2\pi[/Tex]. [Hint: Make a substitution to express the equation only in terms of cosine.]

Q9. Use identities to solve exactly the trigonometric equation over the interval [Tex]0≤x<2π[/Tex].

[Tex]\cos{x}\cos{2x}+\sin{x}\sin{2x}=\frac{\sqrt3}{2}[/Tex]

Q10. Solve the equation exactly using an identity: [Tex]3\cos{\theta}+3=2\sin{2\theta}, 0≤θ<2π[/Tex].

FAQs on Analytic Trigonometry

What is Analytic Trigonometry?

Analytic Trigonometry involves the application of algebraic methods to trigonometric functions and identities to solve equations and analyze properties of trigonometric functions.

How do trigonometric identities help in solving equations?

Trigonometric identities simplify expressions and make it easier to solve trigonometric equations by transforming them into more manageable forms.

What are inverse trigonometric functions used for?

Inverse trigonometric functions are used to find the angles corresponding to given trigonometric function values.

What are the fundamental trigonometric functions?

The fundamental trigonometric functions are sine, cosine, tangent, and their reciprocals—cosecant, secant, and cotangent.