Find Horizontal Tangent Pts for y = 9sin(x)cos(x)

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SUMMARY

The discussion focuses on finding horizontal tangent points for the function y = 9sin(x)cos(x). The derivative is calculated as y' = 9cos²(x) - 9sin²(x). To find horizontal tangents, the equation 9cos²(x) - 9sin²(x) = 0 must be solved, simplifying to cos²(x) - sin²(x) = 0. This leads to the conclusion that horizontal tangent points occur when cos²(x) equals sin²(x).

PREREQUISITES
  • Understanding of trigonometric identities, specifically sin²(x) and cos²(x).
  • Knowledge of calculus, particularly derivatives and their applications in finding tangent lines.
  • Familiarity with solving equations involving trigonometric functions.
  • Basic graphing skills to visualize the function and its tangents.
NEXT STEPS
  • Study the trigonometric identity cos²(x) + sin²(x) = 1 to reinforce understanding of the relationship between sine and cosine.
  • Learn how to apply the first derivative test to determine local maxima and minima in functions.
  • Explore the concept of critical points in calculus and their significance in graph analysis.
  • Practice solving trigonometric equations to gain proficiency in finding points of interest on graphs.
USEFUL FOR

Students studying calculus, particularly those focusing on derivatives and tangent lines, as well as educators seeking to explain the concept of horizontal tangents in trigonometric functions.

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Homework Statement



y = 9sin(x)cos(x)

Find all points where tangent line is horizontal.

The Attempt at a Solution



I get y' = 9cos^2x - 9sin^2X

I plug in zero for the slope and get 9 but I'm stumped after that. How can I get all the horizontal tangent line points?
 
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Physics1 said:

Homework Statement



y = 9sin(x)cos(x)

Find all points where tangent line is horizontal.

The Attempt at a Solution



I get y' = 9cos^2x - 9sin^2x
I plug in zero for the slope and get 9 but I'm stumped after that.
How?
How can I get all the horizontal tangent line points?
You do exactly what you say to do; that is, find the points where the derivative of the graph is zero. It boils down to solving cos^2x-sin^2x=0. Can you do this?

[Hint: cos^2x=1-sin^2x]
 

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