The Purplemath ForumsHelping students gain understanding and self-confidence in algebra powered by FreeFind

Quadrants and Angles (page 3 of 3)

Sections: Introduction, Worked Examples (and Sign Chart), More Examples

• Find the values of the remaining trigometric ratios if
and
sin(α) < 0.
• The tangent is positive in QI and in QIII. The sine ratio is negative in QIII and QIV. The overlap is QIII, so α must terminate in the third quadrant. In the third quadrant, each of x and y is negative, so the numerator and denominator of the tangent ratio y/x are both negative.

 I'll draw what I've got so far: The Pythagorean Theorem gives me:
 The hypotenuse r is always positive, so r = +4 and now my picture becomes:

Then the other ratios are:

The answers for the cosecant and cotangent can be expressed in either of two ways, depending on whether your particular text book still cares whether you leave radicals in the denominator. Your calculus book probably won't care; your algebra book definitely did care; trig books vary. If you're not sure what is the protocol for your class, ask your instructor.

• The equation 5x + 3y = 0, x < 0, is the equation of the terminal side of an angle alpha. Find the values of the six trigonometric ratios for this angle.
• This is an unusual sort of exercise, but I can use what I've learned in algebra to pick it apart. First, I'll solve for "y=" to get the equation y = –(5/3)x. From what I remember of graphing, this is an decreasing line through the origin, so it passes through QII and QIV. Since they gave me the restriction that x is negative, then the angle alpha must end in QII.

 So I've got this much so far:
 Copyright © Elizabeth Stapel 2010-2011 All Rights Reserved Dropping the perpendicular to create the right triangle, I can label the side lengths from the point, and also label the angle, to get:

The height is y = 5 and the base is x = –3, so the hypotenuse is given by:

 52 + (–3)2 = r2 25 + 9 = r2 34 = r2 Since r is always positive, then , so my triangle is:

Then the six trig ratios are:

<< Previous  Top  |  1 | 2 | 3  |  Return to Index

Purplemath:
Printing pages
School licensing

Reviews of
Internet Sites:
Free Help
Practice
Et Cetera

The "Homework
Guidelines"

Study Skills Survey

Tutoring from Purplemath
Find a local math tutor

This lesson may be printed out for your personal use.