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Trigonometric Functions – Formulas, Definition With Examples

Jo-ann Caballes

Updated on January 8, 2024

Welcome to the Brighterly family, where we bring learning to life and illuminate the path of knowledge with joy and simplicity. In today’s journey, we’re going to embark on an exciting adventure through the fascinating world of Trigonometric Functions. These mathematical tools are integral to our daily lives and are widely used in various fields such as physics, engineering, computer science, and even music!

Trigonometric functions, while sounding complicated, are fundamentally about relationships – relationships between angles and sides of a triangle. Once you understand these relationships, you will see how trigonometry permeates various facets of our lives – from determining the height of a mountain using simple measurements to unraveling the secrets of oscillations in physics and music.

Trigonometric Functions

For many, the phrase Trigonometric Functions may seem intimidating. Yet, these concepts are fundamental pillars in the realm of mathematics, forming the foundation of fields like geometry, physics, engineering, and computer science. By breaking these terms down into bite-sized pieces, our understanding of these principles will gradually grow, making them not only manageable but exciting to explore. With the aid of real-world examples and interactive exercises, we will navigate the intricacies of this essential mathematical tool together on Brighterly.

What Are Trigonometric Functions?

Trigonometric functions are a group of functions that link the angles and sides of a triangle. They are fundamentally associated with the measurement of angles and periodic phenomena. These mathematical superheroes come in six primary forms: sine, cosine, tangent, cosecant, secant, and cotangent. Each of these possesses unique relationships and properties, which are vital in solving problems involving angles, real-world physics, and even describing cyclical patterns such as sound and light waves.

Definition of Sine, Cosine, and Tangent

At the heart of trigonometry, we encounter the primary trigonometric functions: sine, cosine, and tangent. Their definitions are rather simple once you get the hang of it. The sine (sin) of an angle in a right triangle is the length of the opposite side divided by the length of the hypotenuse. The cosine (cos) is the length of the adjacent side divided by the hypotenuse. Lastly, the tangent (tan) is simply the sine divided by the cosine, or equivalently, the length of the opposite side divided by the adjacent side.

Definition of Cosecant, Secant, and Cotangent

Complementing the primary functions are the cosecant, secant, and cotangent. The cosecant (csc) of an angle is the reciprocal of the sine. Therefore, it is the length of the hypotenuse divided by the length of the opposite side. The secant (sec) is the reciprocal of the cosine, i.e., the length of the hypotenuse divided by the adjacent side. The cotangent (cot) is the reciprocal of the tangent, which is the length of the adjacent side divided by the length of the opposite side.

Properties of Trigonometric Functions

Trigonometric functions are not merely defined; they also possess distinct properties. These properties range from basic ones like domain and range to complex periodic behaviors and symmetries. They all play crucial roles in helping us solve an array of mathematical problems and make sense of various scientific phenomena.

Properties of Sine, Cosine, and Tangent

The properties of sine, cosine, and tangent further illuminate their unique characteristics. These three functions are periodic, with the sine and cosine functions having a period of 2π (or 360 degrees), and the tangent having a period of π (or 180 degrees). Moreover, they exhibit symmetry: sine is odd, cosine is even, and tangent is odd. These properties allow us to navigate their behaviors across different quadrants in the Cartesian plane.

Properties of Cosecant, Secant, and Cotangent

Much like their primary counterparts, the properties of cosecant, secant, and cotangent are equally compelling. They are also periodic functions, with cosecant and secant sharing the same period as sine and cosine respectively, and cotangent having the same period as tangent. As for symmetry, cosecant and cotangent are odd, while secant is even. Understanding these properties enables us to anticipate the behavior of these functions in diverse mathematical settings.

Difference Between Primary and Reciprocal Trigonometric Functions

While the primary and reciprocal trigonometric functions might seem similar, the distinction is clear when delving deeper. As we’ve discovered, the primary functions (sine, cosine, tangent) are derived directly from the sides of a right triangle, while the reciprocal functions (cosecant, secant, cotangent) are just that – reciprocals of the primary functions. This understanding helps bridge gaps in mathematical concepts, illustrating the interconnected nature of mathematical phenomena.

Formulas of Trigonometric Functions

Moving beyond definitions and properties, let’s dive into the world of trigonometric formulas. These formulas provide a roadmap for solving complex problems. They allow us to simplify equations, evaluate function values, and conduct a range of operations more efficiently. For instance, they’re crucial in expressing one function in terms of another, determining the exact values of functions, and identifying the solutions to trigonometric equations.

Writing Formulas of Sine, Cosine, and Tangent

Formulating the sine, cosine, and tangent formulas may seem daunting initially, but they’re more straightforward than they appear. For instance, the Pythagorean identity is a key formula connecting sine and cosine: sin²θ + cos²θ = 1. For tangent, an important formula is tanθ = sinθ / cosθ. There are also multiple-angle formulas and addition-subtraction formulas, which enable us to express these functions in various contexts.

Writing Formulas of Cosecant, Secant, and Cotangent

The cosecant, secant, and cotangent formulas follow a similar pattern. Each of them can be written in terms of sine, cosine, and tangent respectively. For instance, we have cscθ = 1 / sinθ, secθ = 1 / cosθ, and cotθ = 1 / tanθ. Other significant formulas include the Pythagorean identities for secant and cosecant, as well as their respective multiple-angle and addition-subtraction formulas.

Practice Problems on Trigonometric Functions

Applying our newfound knowledge to practice problems is the secret sauce for enhancing our understanding. Conquering a diverse range of problems not only boosts our expertise in handling trigonometric functions, but it also fuels our creativity in problem-solving. So, it’s time to roll up our sleeves, summon our curiosity, and dive into the fascinating world of trigonometry with these practice problems.

Let’s start with some basics:

Problem 1: In a right-angled triangle, if the length of the hypotenuse is 10 units and one of the angles (θ) is 30°, find the values of sinθ, cosθ, and tanθ.

Solution 1: We know that in a right-angled triangle, sinθ = opposite/hypotenuse, cosθ = adjacent/hypotenuse, and tanθ = opposite/adjacent. By using the special properties of a 30°-60°-90° triangle, we find that the length of the opposite side is 5 units and the length of the adjacent side is 5√3 units. Thus, sin30° = 5/10 = 0.5, cos30° = 5√3/10 = √3/2, and tan30° = 5/(5√3) = 1/√3.

Problem 2: What are the values of cosec30°, sec30°, and cot30°?

Solution 2: Cosecant, secant, and cotangent are the reciprocals of sine, cosine, and tangent respectively. So, cosec30° = 1/sin30° = 1/0.5 = 2, sec30° = 1/cos30° = 2/√3, and cot30° = 1/tan30° = √3.

Next, let’s explore problems involving trigonometric formulas:

Problem 3: Prove the Pythagorean identity, sin²θ + cos²θ = 1, for θ = 45°.

Solution 3: Substituting the exact values of sin45° and cos45°, which are 1/√2 or √2/2, into the formula gives us ((√2/2)² + (√2/2)²) = (1/2 + 1/2) = 1.

Problem 4: Using the formula for tanθ = sinθ / cosθ, find the value of tan45°.

Solution 4: From the definitions of sine and cosine, we know that sin45° = cos45° = √2/2. Thus, substituting these values into the formula for tanθ gives us tan45° = (√2/2) / (√2/2) = 1.

Conclusion

As we conclude our enriching expedition through Trigonometric Functions, it’s heartening to see how far we’ve come! We’ve braved the waters of complex terminologies, navigated the winds of numerous properties, and discovered the treasure trove of formulas. More importantly, through this journey with Brighterly, we’ve made these concepts our own by applying them to practical problems.

However, learning, much like any voyage, does not stop. Each concept is a stepping stone to another, and with each step, we become more proficient, capable, and confident. As we continue to explore, remember that trigonometry is more than just functions and formulas. It’s a language that nature speaks, and as we become more fluent in it, we will be able to comprehend the profound dialogue that nature is engaged in.

So, keep practicing, keep exploring, and keep illuminating the path of learning. And remember, at Brighterly, we’re here to make your learning journey brighter and merrier!

Frequently Asked Questions on Trigonometric Functions

What are the real-world applications of trigonometric functions?

Trigonometric functions have a vast array of applications in real life. They are used in physics to study phenomena such as waves and oscillations. Engineers use them in fields like signal processing and electrical circuit design. In computer graphics, they help in creating visuals and animations. Trigonometry is also used in geology to calculate plate movement, in astronomy to calculate distances and trajectories, and even in music theory to describe sound waves.

Are trigonometric functions and trigonometric ratios the same?

Yes, in a way, they are. Trigonometric ratios refer to the ratios of different sides of a right triangle, and they form the basis of trigonometric functions. Sine, cosine, and tangent are often first introduced as ratios in a right triangle, but when we extend these concepts to include angles of any size, even beyond 360°, and negative angles, they become trigonometric functions.

Why are there six trigonometric functions? Isn’t it enough to have sine, cosine, and tangent?

The primary trigonometric functions, sine, cosine, and tangent, do provide a lot of information. However, their reciprocal functions – cosecant, secant, and cotangent, give us additional tools for solving problems. Certain problems become easier when we use these reciprocal functions. For example, the law of sines and law of cosines involve the use of sine and cosine, but in some situations, using cosecant or secant might simplify the calculations.

At Brighterly, we believe in empowering our learners with comprehensive and easily accessible information. Keep your curiosity alive and keep learning with us!

Jo-ann Caballes

5 articles

As a seasoned educator with a Bachelor’s in Secondary Education and over three years of experience, I specialize in making mathematics accessible to students of all backgrounds through Brighterly. My expertise extends beyond teaching; I blog about innovative educational strategies and have a keen interest in child psychology and curriculum development. My approach is shaped by a belief in practical, real-life application of math, making learning both impactful and enjoyable.