<< Chapter < Page Chapter >> Page >

When working with right triangles, keep in mind that the same rules apply regardless of the orientation of the triangle. In fact, we can evaluate the six trigonometric functions of either of the two acute angles in the triangle in [link] . The side opposite one acute angle is the side adjacent to the other acute angle, and vice versa.

Right triangle with angles alpha and beta. Sides are labeled hypotenuse, adjacent to alpha/opposite to beta, and adjacent to beta/opposite alpha.
The side adjacent to one angle is opposite the other angle.

Many problems ask for all six trigonometric functions for a given angle in a triangle. A possible strategy to use is to find the sine, cosine, and tangent of the angles first. Then, find the other trigonometric functions easily using the reciprocals.

Given the side lengths of a right triangle, evaluate the six trigonometric functions of one of the acute angles.

  1. If needed, draw the right triangle and label the angle provided.
  2. Identify the angle, the adjacent side, the side opposite the angle, and the hypotenuse of the right triangle.
  3. Find the required function:
    • sine as the ratio of the opposite side to the hypotenuse
    • cosine as the ratio of the adjacent side to the hypotenuse
    • tangent as the ratio of the opposite side to the adjacent side
    • secant as the ratio of the hypotenuse to the adjacent side
    • cosecant as the ratio of the hypotenuse to the opposite side
    • cotangent as the ratio of the adjacent side to the opposite side

Evaluating trigonometric functions of angles not in standard position

Using the triangle shown in [link] , evaluate sin α , cos α , tan α , sec α , csc α , and cot α .

Right triangle with sides of 3, 4, and 5. Angle alpha is also labeled which is opposite the side labeled 4.
sin  α = opposite  α hypotenuse = 4 5 cos  α = adjacent to  α hypotenuse = 3 5 tan  α = opposite  α adjacent to  α = 4 3 sec  α = hypotenuse adjacent to  α = 5 3 csc  α = hypotenuse opposite  α = 5 4 cot  α = adjacent to  α opposite  α = 3 4
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Using the triangle shown in [link] ,evaluate sin t , cos t , tan t , sec t , csc t , and cot t .

Right triangle with sides 33, 56, and 65. Angle t is also labeled which is opposite to the side labeled 33.

sin  t = 33 65 , cos  t = 56 65 , tan  t = 33 56 , sec  t = 65 56 , csc  t = 65 33 , cot  t = 56 33

Got questions? Get instant answers now!

Finding trigonometric functions of special angles using side lengths

It is helpful to evaluate the trigonometric functions as they relate to the special angles—multiples of 30° , 60° , and 45° . Remember, however, that when dealing with right triangles, we are limited to angles between  and 90° .

Suppose we have a 30° , 60° , 90° triangle, which can also be described as a π 6 , π 3 , π 2 triangle. The sides have lengths in the relation s , s 3 , 2 s . The sides of a 45° , 45° , 90° triangle, which can also be described as a π 4 , π 4 , π 2 triangle, have lengths in the relation s , s , 2 s . These relations are shown in [link] .

Two side-by-side graphs of circles with inscribed angles. First circle has angle of pi/3 inscribed, radius of 2s, base of length s and height of length . Second circle has angle of pi/4 inscribed with radius , base of length s and height of length s.
Side lengths of special triangles

We can then use the ratios of the side lengths to evaluate trigonometric functions of special angles.

Given trigonometric functions of a special angle, evaluate using side lengths.

  1. Use the side lengths shown in [link] for the special angle you wish to evaluate.
  2. Use the ratio of side lengths appropriate to the function you wish to evaluate.

Evaluating trigonometric functions of special angles using side lengths

Find the exact value of the trigonometric functions of π 3 , using side lengths.

sin ( π 3 ) = opp hyp = 3 s 2 s = 3 2 cos ( π 3 ) = adj hyp = s 2 s = 1 2 tan ( π 3 ) = opp adj = 3 s s = 3 sec ( π 3 ) = hyp adj = 2 s s = 2 csc ( π 3 ) = hyp opp = 2 s 3 s = 2 3 = 2 3 3 cot ( π 3 ) = adj opp = s 3 s = 1 3 = 3 3
Got questions? Get instant answers now!
Got questions? Get instant answers now!

Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
how can we solve this problem
Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
what is the answer to dividing negative index
Morosi Reply
In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
Shivam Reply
give me the waec 2019 questions
Aaron Reply
Practice Key Terms 6

Get the best Algebra and trigonometry course in your pocket!





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