<< Chapter < Page Chapter >> Page >

The point ( 2 2 , 2 2 ) is on the unit circle, as shown in [link] . Find sin t , cos t , tan t , sec t , csc t , and cot t .

This is an image of a graph of circle with angle of t inscribed with radius 1. Point of (square root of 2 over 2, negative square root of 2 over 2) is at intersection of terminal side of angle and edge of circle.

sin t = 2 2 , cos t = 2 2 , tan t = 1 , s e c t = 2 , csc t = 2 , cot t = 1

Got questions? Get instant answers now!

Finding the trigonometric functions of an angle

Find sin t , cos t , tan t , sec t , csc t , and cot t . when t = π 6 .

We have previously used the properties of equilateral triangles to demonstrate that sin π 6 = 1 2 and cos π 6 = 3 2 . We can use these values and the definitions of tangent, secant, cosecant, and cotangent as functions of sine and cosine to find the remaining function values.

tan π 6 = sin π 6 cos π 6 = 1 2 3 2 = 1 3 = 3 3 sec π 6 = 1 cos π 6 = 1 3 2 = 2 3 = 2 3 3 csc π 6 = 1 sin π 6 = 1 1 2 = 2 cot π 6 = cos π 6 sin π 6 = 3 2 1 2 = 3
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Find sin t , cos t , tan t , sec t , csc t , and cot t . when t = π 3 .

sin π 3 = 3 2 cos π 3 = 1 2 tan π 3 = 3 sec π 3 = 2 csc π 3 = 2 3 3 cot π 3 = 3 3

Got questions? Get instant answers now!

Because we know the sine and cosine values for the common first-quadrant angles, we can find the other function values for those angles as well by setting x equal to the cosine and y equal to the sine and then using the definitions of tangent, secant, cosecant, and cotangent. The results are shown in [link] .

Angle 0 π 6 , or 30° π 4 , or 45° π 3 , or 60° π 2 , or 90°
Cosine 1 3 2 2 2 1 2 0
Sine 0 1 2 2 2 3 2 1
Tangent 0 3 3 1 3 Undefined
Secant 1 2 3 3 2 2 Undefined
Cosecant Undefined 2 2 2 3 3 1
Cotangent Undefined 3 1 3 3 0

Using reference angles to evaluate tangent, secant, cosecant, and cotangent

We can evaluate trigonometric functions of angles outside the first quadrant using reference angles as we have already done with the sine and cosine functions. The procedure is the same: Find the reference angle    formed by the terminal side of the given angle with the horizontal axis. The trigonometric function values for the original angle will be the same as those for the reference angle, except for the positive or negative sign, which is determined by x - and y -values in the original quadrant. [link] shows which functions are positive in which quadrant.

To help remember which of the six trigonometric functions are positive in each quadrant, we can use the mnemonic phrase “A Smart Trig Class.” Each of the four words in the phrase corresponds to one of the four quadrants, starting with quadrant I and rotating counterclockwise. In quadrant I, which is “ A ,” a ll of the six trigonometric functions are positive. In quadrant II, “ S mart,” only s ine and its reciprocal function, cosecant, are positive. In quadrant III, “ T rig,” only t angent and its reciprocal function, cotangent, are positive. Finally, in quadrant IV, “ C lass,” only c osine and its reciprocal function, secant, are positive.

This image is a graph of circle with each quadrant labeled. Under quadrant I, labels for sin t, cos t, tan t, sec t, csc t, and cot t. Under quadrant II, labels for sin t and csc t. Under quadrant III, labels for tan t and cot t. Under quadrant IV, labels for cos t, sec t.
The trigonometric functions are each listed in the quadrants in which they are positive.

Given an angle not in the first quadrant, use reference angles to find all six trigonometric functions.

  1. Measure the angle formed by the terminal side of the given angle and the horizontal axis. This is the reference angle.
  2. Evaluate the function at the reference angle.
  3. Observe the quadrant where the terminal side of the original angle is located. Based on the quadrant, determine whether the output is positive or negative.

Using reference angles to find trigonometric functions

Use reference angles to find all six trigonometric functions of 5 π 6 .

The angle between this angle’s terminal side and the x -axis is π 6 , so that is the reference angle. Since 5 π 6 is in the third quadrant, where both x and y are negative, cosine, sine, secant, and cosecant will be negative, while tangent and cotangent will be positive.

cos ( 5 π 6 ) = 3 2 , sin ( 5 π 6 ) = 1 2 , tan ( 5 π 6 ) = 3 3 , sec ( 5 π 6 ) = 2 3 3 , csc ( 5 π 6 ) = −2 , cot ( 5 π 6 ) = 3

Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

Why is b in the answer
Dahsolar Reply
how do you work it out?
Brad Reply
answer
Ernest
heheheehe
Nitin
(Pcos∅+qsin∅)/(pcos∅-psin∅)
John Reply
how to do that?
Rosemary Reply
what is it about?
Amoah
how to answer the activity
Chabelita Reply
how to solve the activity
Chabelita
solve for X,,4^X-6(2^)-16=0
Alieu Reply
x4xminus 2
Lominate
sobhan Singh jina uniwarcity tignomatry ka long answers tile questions
harish Reply
t he silly nut company makes two mixtures of nuts: mixture a and mixture b. a pound of mixture a contains 12 oz of peanuts, 3 oz of almonds and 1 oz of cashews and sells for $4. a pound of mixture b contains 12 oz of peanuts, 2 oz of almonds and 2 oz of cashews and sells for $5. the company has 1080
ZAHRO Reply
If  , , are the roots of the equation 3 2 0, x px qx r     Find the value of 1  .
Swetha Reply
Parts of a pole were painted red, blue and yellow. 3/5 of the pole was red and 7/8 was painted blue. What part was painted yellow?
Patrick Reply
Parts of the pole was painted red, blue and yellow. 3 /5 of the pole was red and 7 /8 was painted blue. What part was painted yellow?
Patrick
how I can simplify algebraic expressions
Katleho Reply
Lairene and Mae are joking that their combined ages equal Sam’s age. If Lairene is twice Mae’s age and Sam is 69 yrs old, what are Lairene’s and Mae’s ages?
Mary Reply
23yrs
Yeboah
lairenea's age is 23yrs
ACKA
hy
Katleho
Ello everyone
Katleho
Laurene is 46 yrs and Mae is 23 is
Solomon
hey people
christopher
age does not matter
christopher
solve for X, 4^x-6(2*)-16=0
Alieu
prove`x^3-3x-2cosA=0 (-π<A<=π
Mayank Reply
create a lesson plan about this lesson
Rose Reply
Excusme but what are you wrot?
Practice Key Terms 6

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Algebra and trigonometry. OpenStax CNX. Nov 14, 2016 Download for free at https://legacy.cnx.org/content/col11758/1.6
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Algebra and trigonometry' conversation and receive update notifications?

Ask