Suppose $x = 2 \tan A$.Rewrite The Expression In Terms Of Sines And Cosines:$\frac{x^2 + 4}{x} = \square$Enter An Algebraic Expression.

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Suppose x=2tan⁡ax = 2 \tan a

Rewriting the Expression in Terms of Sines and Cosines

In trigonometry, the tangent function is defined as the ratio of the sine and cosine functions. Given the expression x=2tan⁡ax = 2 \tan a, we can rewrite it in terms of sines and cosines by using the definition of the tangent function.

The Tangent Function

The tangent function is defined as:

tan⁡a=sin⁡acos⁡a\tan a = \frac{\sin a}{\cos a}

Using this definition, we can rewrite the expression x=2tan⁡ax = 2 \tan a as:

x=2sin⁡acos⁡ax = 2 \frac{\sin a}{\cos a}

Simplifying the Expression

To simplify the expression, we can multiply both sides by cos⁡a\cos a to get:

xcos⁡a=2sin⁡ax \cos a = 2 \sin a

Rewriting the Expression in Terms of Sines and Cosines

Now, we can rewrite the expression x2+4x\frac{x^2 + 4}{x} in terms of sines and cosines by substituting the expression for xx in terms of sin⁡a\sin a and cos⁡a\cos a.

x2+4x=(2sin⁡a)2+42sin⁡a\frac{x^2 + 4}{x} = \frac{(2 \sin a)^2 + 4}{2 \sin a}

Expanding the Expression

To expand the expression, we can square the numerator and simplify:

(2sin⁡a)2+42sin⁡a=4sin⁡2a+42sin⁡a\frac{(2 \sin a)^2 + 4}{2 \sin a} = \frac{4 \sin^2 a + 4}{2 \sin a}

Simplifying the Expression

To simplify the expression, we can divide both sides by 2sin⁡a2 \sin a to get:

4sin⁡2a+42sin⁡a=2sin⁡a+2sin⁡a\frac{4 \sin^2 a + 4}{2 \sin a} = 2 \sin a + \frac{2}{\sin a}

Rewriting the Expression in Terms of Sines and Cosines

Now, we can rewrite the expression 2sin⁡a+2sin⁡a2 \sin a + \frac{2}{\sin a} in terms of sines and cosines by using the identity sin⁡2a+cos⁡2a=1\sin^2 a + \cos^2 a = 1.

2sin⁡a+2sin⁡a=2sin⁡a+2cos⁡a2 \sin a + \frac{2}{\sin a} = 2 \sin a + 2 \cos a

Simplifying the Expression

To simplify the expression, we can combine like terms:

2sin⁡a+2cos⁡a=2sec⁡a+2tan⁡a2 \sin a + 2 \cos a = \boxed{2 \sec a + 2 \tan a}

Conclusion

In this article, we have rewritten the expression x2+4x\frac{x^2 + 4}{x} in terms of sines and cosines by using the definition of the tangent function and the identity sin⁡2a+cos⁡2a=1\sin^2 a + \cos^2 a = 1. We have also simplified the expression to get the final result of 2sec⁡a+2tan⁡a2 \sec a + 2 \tan a.

References

  • [1] "Trigonometry" by Michael Corral
  • [2] "Calculus" by Michael Spivak

Table of Contents

Q: What is the definition of the tangent function?

A: The tangent function is defined as the ratio of the sine and cosine functions. It is given by:

tan⁡a=sin⁡acos⁡a\tan a = \frac{\sin a}{\cos a}

Q: How can we rewrite the expression x=2tan⁡ax = 2 \tan a in terms of sines and cosines?

A: We can rewrite the expression x=2tan⁡ax = 2 \tan a in terms of sines and cosines by using the definition of the tangent function. We get:

x=2sin⁡acos⁡ax = 2 \frac{\sin a}{\cos a}

Q: How can we simplify the expression x=2sin⁡acos⁡ax = 2 \frac{\sin a}{\cos a}?

A: We can simplify the expression by multiplying both sides by cos⁡a\cos a to get:

xcos⁡a=2sin⁡ax \cos a = 2 \sin a

Q: How can we rewrite the expression x2+4x\frac{x^2 + 4}{x} in terms of sines and cosines?

A: We can rewrite the expression x2+4x\frac{x^2 + 4}{x} in terms of sines and cosines by substituting the expression for xx in terms of sin⁡a\sin a and cos⁡a\cos a. We get:

x2+4x=(2sin⁡a)2+42sin⁡a\frac{x^2 + 4}{x} = \frac{(2 \sin a)^2 + 4}{2 \sin a}

Q: How can we expand the expression (2sin⁡a)2+42sin⁡a\frac{(2 \sin a)^2 + 4}{2 \sin a}?

A: We can expand the expression by squaring the numerator and simplifying:

(2sin⁡a)2+42sin⁡a=4sin⁡2a+42sin⁡a\frac{(2 \sin a)^2 + 4}{2 \sin a} = \frac{4 \sin^2 a + 4}{2 \sin a}

Q: How can we simplify the expression 4sin⁡2a+42sin⁡a\frac{4 \sin^2 a + 4}{2 \sin a}?

A: We can simplify the expression by dividing both sides by 2sin⁡a2 \sin a to get:

4sin⁡2a+42sin⁡a=2sin⁡a+2sin⁡a\frac{4 \sin^2 a + 4}{2 \sin a} = 2 \sin a + \frac{2}{\sin a}

Q: How can we rewrite the expression 2sin⁡a+2sin⁡a2 \sin a + \frac{2}{\sin a} in terms of sines and cosines?

A: We can rewrite the expression 2sin⁡a+2sin⁡a2 \sin a + \frac{2}{\sin a} in terms of sines and cosines by using the identity sin⁡2a+cos⁡2a=1\sin^2 a + \cos^2 a = 1. We get:

2sin⁡a+2sin⁡a=2sin⁡a+2cos⁡a2 \sin a + \frac{2}{\sin a} = 2 \sin a + 2 \cos a

Q: What is the final result of rewriting the expression x2+4x\frac{x^2 + 4}{x} in terms of sines and cosines?

A: The final result of rewriting the expression x2+4x\frac{x^2 + 4}{x} in terms of sines and cosines is:

2sec⁡a+2tan⁡a2 \sec a + 2 \tan a

Q: What are some common identities used in trigonometry?

A: Some common identities used in trigonometry include:

  • sin⁡2a+cos⁡2a=1\sin^2 a + \cos^2 a = 1
  • tan⁡a=sin⁡acos⁡a\tan a = \frac{\sin a}{\cos a}
  • sec⁡a=1cos⁡a\sec a = \frac{1}{\cos a}
  • csc⁡a=1sin⁡a\csc a = \frac{1}{\sin a}

Q: How can we use these identities to simplify trigonometric expressions?

A: We can use these identities to simplify trigonometric expressions by substituting the expressions for the trigonometric functions in terms of the sine and cosine functions.

Q: What are some common applications of trigonometry?

A: Some common applications of trigonometry include:

  • Navigation: Trigonometry is used in navigation to calculate distances and angles between locations.
  • Physics: Trigonometry is used in physics to describe the motion of objects and the behavior of waves.
  • Engineering: Trigonometry is used in engineering to design and build structures such as bridges and buildings.
  • Computer Science: Trigonometry is used in computer science to create 3D graphics and animations.

Conclusion

In this article, we have answered some common questions about rewriting the expression x2+4x\frac{x^2 + 4}{x} in terms of sines and cosines. We have also discussed some common identities used in trigonometry and their applications.