Trigonometry might sound intimidating at first, but it’s actually one of the coolest tools in math! It’s all about the connection between the sides and angles of triangles, using ratios like sine, cosine, and tangent. But here’s the best part—whether you're taking the SAT, ACT, PSAT/NMSQT, GED, GRE, GMAT, AP exams, PERT, Accuplacer, or even prepping for the MCAT, trigonometry pops up everywhere. It helps you solve tricky triangle problems fast, and it's used in fields like engineering, architecture, astronomy, and physics. In this article, we’ll break down the key trig ratios, identities, standard angles, and show you exactly how they’re applied, so you’ll feel confident and test-ready in no time!

List of Topics under Trigonometry

In the introduction, we read about the trigonometry basics, let us start our discussion with a quick overview of the different topics that are counted under the subject.

Trigonometric Ratios

To understand the concept of trigonometric ratios let us consider the below figure:

The figure depicts a triangle ΔABC, right-angled at B and angle A = θ. The side which is opposite to θ is the perpendicular or opposite side. The side that is opposite to angle B is termed the hypotenuse and the side opposite to angle C is known as the base or adjacent side. Now let’s try to write the formula for the sin, cos, tan, etc.

Sine Function(sin)= perpendicular/ Hypotenuse=BC/AC

Cosine Function(cos)= base / Hypotenuse=AB/AC

Tangent Function(tan)=perpendicular / base=BC/AB

Cosecant Function(cosec)=Hypotenuse / perpendicular=AC/BC

Secant Function(sec)=Hypotenuse / base =AC/AB

Cotangent Function(cot)=base/ perpendicular=AB/BC

If you are reading Trigonometric theories, also read about Three Dimensional Geometry here.

Important Trigonometric Identities

The list of all the important trigonometric identities in mathematics is given below:

Fundamental Identities

The basic fundamental identities are as follows:

• sin²x + cos²x = 1
• 1 +tan²x= sec²x
• 1 + cot²x = cosec²x

Sum, Difference, and Products of Two Angles Identities

The various sum, difference, and products of two angles identities are listed below:

• sin (A ± B) = sin A cos B ± cos A sin B
• cos (A ± B) = cos A cos B ∓ sin A sin B
•  2 sin A cos B = sin (A + B) + sin (A – B)
• 2 cos A sin B = sin (A + B) – sin (A – B)
• 2 cos A cos B = cos (A + B) + cos (A – B)
• 2 sin A sin B = cos (A – B) – cos (A + B)
• \(\tan\left(a\ +b\right)=\frac{\left(\tan a\ +\tan b\right)}{1-\tan a\tan b}\)
• \(\tan\left(a\ -b\right)=\frac{\left(\tan a\ -\tan b\right)}{1+\tan a\tan b}\)
• \(\cot\left(a\ +b\right)=\frac{\left(\cot a\cot b-1\right)}{\cot b+\cot a}\)
• \(\cot\left(a\ -b\right)=\frac{\left(\cot a\cot b+1\right)}{\cot b-\cot a}\)
• \(\sin c+\sin d=2\sin\left(\frac{c+d}{2}\right)\cos\left(\frac{c-d}{2}\right)\)
• \(\sin c-\sin d=2\sin\left(\frac{c-d}{2}\right)\cos\left(\frac{c+d}{2}\right)\)

Multiple Angle Identities

The different multiple angle identities are listed below:

• sin(2A) = 2sin(A) cos(A) = [2tan A/(1+tan²A)]
• cos 2A = cos²A– sin²A= [(1-tan²A)/(1+tan²A)]
• tan 2A = 2 tan A / (1 – tan²A)
• cot 2A = (cot²A – 1) / 2 cot A
• cos(2A) = 2cos²A−1 = 1–2sin²A
• \(\tan3A=[3\tan A-\tan^3A]/[1-3\tan^2A]\)
• \(\cot3A=(3\cot x-Cot^3x)/(1-3Cot^2x)\)
• \(Sin3A=3\sin A{-4}\sin^3A\)
• \(\cos3A=4\cos^3A-3\cos A\)

Trigonometric Functions in Different Quadrants

Now that we know about the different trigonometric functions along with the related formulas, below is a summary of the sign of different functions in the different quadrants in a tabular format.

Trigonometric Values of some Standard Angles

In the previous header, we read about signs of the six functions in different functions. The below trigonometry table shows the different values of the functions in terms of degree.

Domain and Range of Trigonometric Functions

The value of θ in sin, cos, tan, cosec, sec and cot denotes the domain of the trigonometric functions and the resultant of the value is the range of the trig functions. The domain and range of of different trigonometric functions are as follows:

Trigonometric Equation with their General Solution

An equation that involves one or more trigonometric ratios of unknown angles is termed a trigonometric equation. The solution to such an equation with variable x, such that it lies in between 0≤x≤2π is said to be a principal solution. On the other hand, if the solution holds an integer say ‘n’ in it, then the equation is called a general solution. The trigonometric equation with the respective general solution of different functions are listed below:

Trigonometric Functions Graphs

Properties like domain, range, periodicity and nature of the different functions can be best studied by employing the trigonometric function graphs. Below is the graph for sin, cos, tan, cosec, sec, and cot.

Applications of Trigonometry

Until now, we have read about the basic concepts of trigonometry with the formula, identities, angles and more. Trigonometry is also used in physics, engineering, satellite navigation, electronics study, architecture, astronomy, oceanography and many such domains. The different real-life applications are listed below:

Real-Life Examples of Trigonometry

• Let us understand the different applications of mathematical trigonometry in various domains.
• In physics subject, the concepts of trigonometry and its formula are used in vectors; to obtain the dot product and cross products, to obtain the components of the vector and so on.
• The trigonometry concepts are even involved in the navigation to point toward a location. The reason is that trigonometric functions can be used to set directions such as east-west or north-south.
• Further, the idea of trigonometry is also used in estimating the height of a building or a mountain, as this can be easily determined using the different trigonometric functions.
• It is also used in the aviation and naval industries.
• Trigonometric functions are also applied to determine the trajectory of a projectile and further estimate the causes of a collision. Also, the concept is applied to locate how an entity falls or at what angle a bullet is shot from a gun. Hence it is used in criminology as well.
• It is also used in marine biology, to measure the depth of sunlight to study the effects on algae that survive on photosynthesis.

Conclusion

Trigonometry isn’t just about triangles—it’s a powerful tool that shows up in everything from physics and engineering to navigation and even crime scene analysis! Whether you're prepping for the SAT, ACT, GED, GRE, or any other big exam, having a strong grip on trig ratios, identities, and formulas will give you a serious edge. Master these concepts now, and you'll be set for both test day success and real-world problem-solving. Keep practicing—you’ve got this!