determining signs of trigonometric functions

determining signs of trigonometric functions

There are at least two

mnemonic devices

for determining the sign a

trigonometric function

at a given angle. They are “all snow tastes cold” and (the more mathematical version) “all students take

calculus

”.

The first in both of these, “all”, indicates that, if an angle lies in the first

quadrant

, then, when any trigonometric function is applied to it, the result is positive.

The second in both of these starts with the letter “s”, which indicates that, if the

terminal ray

of an angle lies in the second quadrant, then the only trigonometric functions that can be applied to it that yield a positive result are

sin

and its

reciprocal

csc

.

The third in both of these starts with the letter “t”, which indicates that, if the terminal ray of an angle lies in the third quadrant, then the only trigonometric functions that can be applied to it that yield a positive result are

tan

and its reciprocal

cot

.

The fourth in both of these starts with the letter “c”, which indicates that, if the terminal ray of an angle lies in the fourth quadrant, then the only trigonometric functions that can be applied to it that yield a positive result are

cos

and its reciprocal

sec

.

Because of how these mnemonic devices work, it is clear that they are in of the

calculator trigonometric functions

.

Below is a picture that illustrates how the mnemonic device “all students take calculus” works:

For angles whose terminal ray lies on the boundary of two quadrants, the matter of determining sign is not as , but it is still possible to do so through use of the mnemonic device. If, for both of the boundary quadrants, the sign the trigonometric function is positive, then the value of the trigonometric function applied to the angle is

1

. If, for both of the boundary quadrants, the sign the trigonometric function is negative, then the value of the trigonometric function applied to the angle is

-

1

. If the sign the trigonometric function is different in the two boundary quadrants, then the value of the trigonometric function applied to the angle is either

0

or undefined.

Example:

Since the terminal ray of

2

⁢

π

3

lies in the second quadrant, we have that

sin

⁡

(

2

⁢

π

3

)

>

0

, and

cos

⁡

(

2

⁢

π

3

)

<

0

.

(

In fact,

sin

⁡

(

2

⁢

π

3

)

=

3

2

and

cos

⁡

(

2

⁢

π

3

)

=

-

1

2

.

)

Example:

Since the terminal ray of

7

⁢

π

2

lies on the boundary of the third and fourth quadrants and, when

csc

is applied to any angle whose terminal ray lies in either the third or fourth quadrant, the result is negative, we have that

csc

⁡

(

7

⁢

π

2

)

=

-

1

.

Title

determining signs of trigonometric functions

Canonical name

DeterminingSignsOfTrigonometricFunctions

Date of creation

2013-03-22 16:06:08

Last modified on

2013-03-22 16:06:08

Owner

Wkbj79 (1863)

Last modified by

Wkbj79 (1863)

Numerical id

11

Author

Wkbj79 (1863)

Entry type

Topic

Classification

msc 51-01

Classification

msc 97D40

Synonym

all snow tastes cold

Synonym

all students take calculus

Related topic

Trigonometry

Related topic

CalculatorTrigonometricFunctions

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