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### Course: Trigonometry>Unit 1

Lesson 7: The reciprocal trigonometric ratios

# Trigonometric ratios review

Review all six trigonometric ratios: sine, cosine, tangent, cotangent, secant, & cosecant.

## What are the trigonometric ratios?

$\mathrm{sin}\left(\mathrm{\angle }A\right)=$$\frac{\text{opposite}}{\text{hypotenuse}}$
$\mathrm{cos}\left(\mathrm{\angle }A\right)=$$\frac{\text{adjacent}}{\text{hypotenuse}}$
$\mathrm{tan}\left(\mathrm{\angle }A\right)=$$\frac{\text{opposite}}{\text{adjacent}}$
$\mathrm{cot}\left(\mathrm{\angle }A\right)=$$\frac{\text{adjacent}}{\text{opposite}}$
$\mathrm{sec}\left(\mathrm{\angle }A\right)=$$\frac{\text{hypotenuse}}{\text{adjacent}}$
$\mathrm{csc}\left(\mathrm{\angle }A\right)=$$\frac{\text{hypotenuse}}{\text{opposite}}$
Want to learn more about sine, cosine, and tangent? Check out this video.

## Practice set 1: sine, cosine, and tangent

Problem 1.1
$\mathrm{sin}\left(\mathrm{\angle }B\right)=$
Use an exact expression.

Want to try more problems like this? Check out this exercise.

## Practice set 2: cotangent, secant, and cosecant

Problem 2.1
$\mathrm{cot}\left(\mathrm{\angle }B\right)=$
Use an exact expression.

Want to try more problems like this? Check out this exercise.

## Want to join the conversation?

• Why aren't the reciprocal functions taught with the normal three?
Are they simply less used or are they harder to teach without sin, cos, and tan?
• Both. They are less used and without the 3 foundational functions, they are a touch harder to teach. We often teach using SOH-CAH-TOA and using a right triangle, so sin/cos/tan are very well known.
• Are cse, sec and cot in a calculator?
• Sometimes. There are also sometimes inverses of all of them, AND hyperbolic versions of all of those!
• what would be some applications for using the inverse functions? BW- they seem more intuitive then the sine, and cosine. Tangent seems more intuitive too.
• Do you mean the "Reciprocal functions" like secant and cosecant. The inverse trigonometric functions (the cyclometric functions) are represented by arcosine, arcsine etc.

Reciprocal functions were used in tables before computer power went up and there are some instances where calculating an inverse of a function is easier than the function.

As to Inverse tringonometric functions they are used to calculate angles.
• Is there an inverse for the reciprocal functions: cosecant, secant, and cotangent?
• Yes, they're arccosecant, arcsecant, and arccotangent.
• Can trigonometry be applied in higher dimensions? if so where?
• Yes, trigonometry can be extended to higher dimensions, and this is often referred to as "multidimensional trigonometry" or "hyperbolic trigonometry" in some contexts. While the basic trigonometric functions like sine, cosine, and tangent are defined in the context of two-dimensional right triangles, they can be generalized to higher dimensions using concepts from linear algebra and vector spaces.

Here are some ways in which trigonometric concepts can be applied in higher dimensions:

Spherical Trigonometry: Spherical trigonometry deals with triangles on the surface of a sphere. It extends the concepts of traditional trigonometry to the three-dimensional space of the sphere. Spherical trigonometry is particularly important in fields such as astronomy, navigation, and geodesy.

Hyperbolic Trigonometry: Hyperbolic trigonometry deals with hyperbolic functions like hyperbolic sine (sinh), hyperbolic cosine (cosh), and hyperbolic tangent (tanh). These functions can be used in various mathematical and scientific contexts, including geometry and physics.

Vector Spaces: In linear algebra and vector calculus, trigonometric functions can be used to describe the relationships between vectors in higher-dimensional spaces. For example, the dot product between two vectors involves the cosine of the angle between them, and this concept extends to higher dimensions.

Geometry: Trigonometric concepts can be used to study angles, distances, and relationships between objects in higher-dimensional Euclidean spaces. They can also be applied to problems involving polyhedra and other geometric figures in higher dimensions.

Engineering and Physics: Trigonometric concepts are used in various engineering and physics applications that involve multidimensional systems, such as wave propagation, oscillations, and vibrations in three-dimensional space or higher.

Computer Graphics and Computer Science: Trigonometry plays a role in computer graphics when dealing with three-dimensional modeling, rotation, and transformations. It is also used in computer science algorithms and data analysis in multidimensional spaces.

In these higher-dimensional contexts, the trigonometric functions are generalized and adapted to work with vectors, matrices, and other mathematical structures. This allows for the analysis and manipulation of data and phenomena in multiple dimensions, making trigonometry a valuable tool in various fields of mathematics and science.
• So there are sine, cosine, tangent, arcsine, arccosine, arctangent, cosecant, secant, and cotangent. My calculator says there also seems to be arcsecant, arccosecant, and arccotangent. Is that correct? Are they called by different names?
• Arcsecant, arccosecant, and arctangent are all inverses of the reciprocal functions.
• What are the hyperbolic trig functions?