How Do Cosine and Sine Relate to Arctan in Trigonometry?

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SUMMARY

The relationship between cosine and sine with arctan in trigonometry is defined by the equations cos(arctan x) = 1 / √(1+x²) and sin(arctan x) = x / √(1+x²). This stems from the properties of a right triangle where one side is 1 and the other side is x, leading to the hypotenuse being √(1+x²). The angle corresponding to arctan(x) is derived from the tangent ratio, which is the ratio of the opposite side to the adjacent side in a right triangle.

PREREQUISITES
  • Understanding of right triangle properties
  • Familiarity with trigonometric functions (sine, cosine, tangent)
  • Knowledge of inverse trigonometric functions, specifically arctan
  • Basic algebraic manipulation skills
NEXT STEPS
  • Study the derivation of trigonometric identities involving arctan
  • Learn about the unit circle and its relation to trigonometric functions
  • Explore the Pythagorean theorem and its applications in trigonometry
  • Investigate the graphical representation of sine, cosine, and tangent functions
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Students studying physics, mathematics enthusiasts, and anyone looking to deepen their understanding of trigonometric relationships and their applications in problem-solving.

rainyrabbit
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Why is cos(arctan x) = 1 / root(1+x^2)

and sin(arctan x) = x / root(1+x^2)?

I would greatly appreciate help. I'm studying for physics and this came up in one of the solutions. Thank you.
 
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Think of a right triangle with sides 1 and x. Now try to imagine the values of:
- The hypotenuse, and
- The sin, cos and tan of the angle next to the "1" side.
 
Well, you could show this algebraically, however:

What is arctan(x)? arctan(x) is that angle whose tan-value is x.

Now, writing x=x/1, remember that tan can be thought of as the ratio between two of the sides in a right-angled triangle.

Identify which sides these are, and see if you can deduce the results for yourself.
 

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