LESSON READ-THROUGH through Dr. Carol JVF Burns (website creator) Follow along with the highlighted message while girlfriend listen!

You are watching: Tangent is positive in quadrants i and iv only

sometimes in problem-solving friend only require the sign (plus or minus) the a trigonometric value—not its size. With 3 already-studied concepts, girlfriend have accessibility to the signs of all the trigonometric functions, in all the quadrants: basic arithmetic facts about signs of quotients:
 $\,\displaystyle\frac\textPOS\textNEG = \textNEG$ $\,\displaystyle\frac\textNEG\textPOS = \textNEG$ $\,\displaystyle\frac\textNEG\textNEG = \textPOS$ (positive divided by negativeis negative) (negative divided by positiveis negative) (negative split by negativeis positive)

## Signs (plus/minus) that Sine and Cosine in all Quadrants

 through definition, $\,\cos \theta\,$ and $\,\sin\theta\,$ give the $\,x\,$ and also $\,y\,$ worths (respectively) of clues on the unit circle,as displayed at right. It follows that: sine is positive in quadrants I and also II: points over the $x$-axis have actually positive $y$-values sine is an unfavorable in quadrants III and IV: points listed below the $x$-axis have an unfavorable $y$-values cosine is hopeful in quadrants I and IV: points to the best of the $y$-axis have actually positive $x$-values cosine is an adverse in quadrants II and also III: points come the left that the $y$-axis have negative $x$-values recall also: The number zero has no sign: $\,0\,$ is no positive and $\,0\,$ is no negative. Zero is neutral! because that example, $\,\cos \frac\pi2 = \cos 90^\circ = 0\,$, hence has no sign. If a number is no defined, climate (of course) it has no sign. Because that example, $\,\tan\frac\pi2 = \tan 90^\circ\,$ is no defined, hence has actually no sign.

## Signs (plus/minus) the Tangent in all Quadrants

 by definition, $\displaystyle\,\tan\theta :=\frac\sin\theta\cos\theta\,$. Therefore: In quadrant I, tangent is positive: $\displaystyle\fracPOSPOS = POS$ In quadrant II, tangent is negative: $\displaystyle\fracPOSNEG = NEG$ In quadrant III, tangent is positive: $\displaystyle\fracNEGNEG = POS$ In quadrant IV, tangent is negative: $\displaystyle\fracNEGPOS = NEG$

## Memory an equipment for the indicators (plus/minus) of Sine, Cosine, and also Tangent in every Quadrants

 The (optimistic!) memory an equipment every STUDENTS take CALCULUS answers the question Where room the (first three) trigonometric functions positive? They"re ALL hopeful in Quadrant I. Only the Sine (the ‘S’ in ‘Students’) is optimistic in Quadrant II. Just the Tangent (the ‘T’ in ‘Take’) is optimistic in Quadrant III. Only the Cosine (the ‘C’ in ‘Calculus’) is optimistic in Quadrant IV. of course, you must remember to begin with words ‘ALL’ in quadrant I, and then proceed counterclockwise. This memory an equipment answers the question: Where room the sine, cosine, and also tangent POSITIVE?

## Signs (plus/minus) the Cotangent, Secant, and also Cosecant in all Quadrants

 The reciprocal of a number retains the sign of the original number. Therefore, in every the quadrants: Cosecant has actually the exact same sign as sine. Secant has actually the same sign together cosine. Cotangent has the same sign together tangent.

grasp the principles from this section by practicing the practice at the bottom that this page. when you"re excellent practicing, relocate on to: Trigonometric worths of one-of-a-kind Angles

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 top top this exercise, you will certainly not crucial in your answer. However, friend can check to watch if her answer is correct. difficulty TYPES: 1 2 3 4 5 6 7 8
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