Solving precalc problem without calculator

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SUMMARY

The problem involves finding sec theta for an angle whose terminal side passes through the point (-2, -3) in standard position. To solve this without a calculator, one effective method is to construct a right triangle using the coordinates, applying the Pythagorean theorem to determine the lengths of the sides. The secant function is defined as the reciprocal of the cosine, thus sec theta can be calculated once the cosine value is determined from the triangle's dimensions. Alternatively, utilizing the circular definition of sine and cosine by drawing a unit circle can simplify the process of finding the corresponding point on the line.

PREREQUISITES
  • Understanding of trigonometric functions, specifically secant and cosine.
  • Familiarity with the Pythagorean theorem.
  • Knowledge of Cartesian coordinate systems.
  • Ability to draw and interpret unit circles.
NEXT STEPS
  • Study the properties and applications of the secant function in trigonometry.
  • Learn how to apply the Pythagorean theorem in various geometric contexts.
  • Explore the concept of unit circles and their role in defining trigonometric functions.
  • Practice solving trigonometric problems involving angles in standard position.
USEFUL FOR

Students studying precalculus, educators teaching trigonometry, and anyone looking to enhance their understanding of trigonometric functions and their applications in geometry.

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The terminal side of an angle theta in standard position passes through (-2,-3). What is sec theta?

How would you solve this problem without a calculator?

Thanks.
 
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One approach to start with is draw this angle on a cartesian coordinate system, maybe including in the form of a triangle; you may need to express something using pythagorean theorem. Remember that 1/(cosine) = secant.
 
you could set up a triangle having (0,0) as one vertex, (-2, -3) as another and (-2, 0) as the third. What are the lengths of the three sides? What is sec of the angle at the origin (which is what you must mean by "theta"- you didn't specify what "theta" was!). Remember that these are "signed" lengths. Distances measured downward or to the left are negative.

Even simpler would be to use the "circular" definition of sine and cosine: Draw a unit circle. Starting at (1, 0) measure around the circumference a distance t. The coordinates of the end point are, by definition, (cos t, sin t). Now, (-2, -3) does not lie on the unit circle but it is easy to find a point on that line that does (use "similar triangles"). What is the x coordinate of that point? What is 1/cosine?
 

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