Tangent and Right Triangles The tangent function is the function defined as tan(θ) =

sin(θ) cos(θ)

The implied domain of the tangent function is every number θ except for 5π those which have cos(θ) = 0 : the numbers . . . , − π2 , π2 , 3π 2 , 2 ,...

Graph of tangent

Period of tangent Tangent is a periodic function with period π, meaning that tan(θ + π) = tan(θ) This follows from Lemma 10 of the previous chapter which stated that cos(θ + π) = − cos(θ)

and

sin(θ + π) = − sin(θ)

Therefore, tan(θ + π) =

sin(θ + π) − sin(θ) sin(θ) = = = tan(θ) cos(θ + π) − cos(θ) cos(θ) 236

Tangent is an odd function Recall that an even function is a function f (x) that has the property that f ( x) = f (x) for every value of x. Examples of even functions include x2 , x4 , x6 , and cos(x). An odd function is a function g(x) that has the property g( x) = g(x) for every value of x. Examples of odd functions include x3 , x5 , x7 , and sin(x). We can add tangent to our list of odd functions. To see why tangent is odd, we’ll use that sine is odd and cosine is even: tan( ✓) =

*

*

*

*

sin( ✓) sin(✓) = = cos( ✓) cos(✓) *

*

* * /4*L

tan(✓)

* S* / Co?€

*

*

*

Trigonometry for right triangles Trigonometry for right triangles

Suppose that we have a right triangle. That is, a triangle one of whose Suppose a right triangle.that That is, a triangle one of whose angles equals call⇡ we We that the have side of the triangle is opposite the right equals call the side of the triangle that is opposite the right 2 . We angle the angles hypotenuse of the triangle. angle the hypotenuse of the triangle. .

kypoten u se.

If we focus our attention on a second angle of the right triangle, an angle that we’ll call 8, then we can label the remaining two sides as either being opposite from 8, orfocus adjacent to 8. We’ll the lengths sides of an angle If we our attention on call a second angle of the the three right triangle, the triangle hyp, opp,call and✓, adj. that we’ll then we can label the remaining two sides as either being opposite from ✓, or adjacent to ✓. We’ll call the lengths of the three sides of the triangle hyp, opp, and adj. See the picture on the following page. 237

We call the side of the angles triangleequals that is opposite We call the the right side of the triangle that is opposite the right se of the triangle. angle the hypotenuse of the triangle. .

hypot€nus€ oppOte.

GkcE. nt

aci

ttention on a second angle of the right anon angle If we focus ourtriangle, attention a second triangle, an angle The following proposition relates the lengths of theangle sidesof of the the right triangle hen we can label the remaining two sides as either being call 8, then that weangle can label thethe remaining two sides as either being shown above to we’ll the measure of the θ using trigonometric functions adjacent to sine, 6. We’ll call the tangent. lengths of or theadjacent three sides opposite from 6, to 6.of We’ll call the lengths of the three sides of cosine, and (e) (co), pp, and adj. the triangle hyp, opp, and adj. Proposition (13). In a right triangle as shown above, the following equations hold:

sin(θ) =

opp hyp

cos(θ) =

s(e)

adj hyp

z

tan(θ) =

opp adj

Proof: We begin with the triangle on the top right of this page, a right CO (e) triangle with its side lengths labelled as instructed above: hyp for the hypotenuse, opp for the side opposite θ, and adj for the remaining side, which is adjacent to θ. The triangle below on the left is the first triangle scaled by 1 the number hyp . That is, all lengths have been divided by the number hyp, but the angles remain unchanged. What we have then is a new right triangle whose hypotenuse has length hyp hyp = 1. e three formulas Then we have the three formulas

=

opp hyp

—

cos(6)

=

adj hyp

—

opp tan(6) sin(6)== adj hyp

cos(6)

—

kyp

216

=

adj hyp

—

SIi

tan(6)

(e))

=

216

a4j

kyp

The picture on the right shows sin(θ) and cos(θ). Notice that the triangles opp adj from the left and right are the same, so sin(θ) = hyp and cos(θ) = hyp . 238

Last, notice that from the definition of tangent we have opp hyp adj hyp

sin(θ) tan(θ) = = cos(θ)

=

opp adj

 Problem. Find sin(8), eos(8), and tan(8) for the angle 8 shown belo Problem. Find sin(θ), cos(θ), and tan(θ) for the angle θ shown below.

5

5

3

3

It

Lj.

Solution. The hypotenuse is the side that is opposite the right angle Solution. The hypotenuse is the side that is opposite the right angle. It has length 5. Of the two remaining sides, the one that is opposite of 8 ha length 5. Of the two remaining sides, the one that is opposite of θ has length 4, and the one that is adjacent to 0 has length 3. Therefore, 4, and the one that is adjacent to θ has length 3. Therefore, O]3J3 4 4 sin(0) opp = sin(θ) = hyp 5 —

—

--

*

*

*

*

adj 3 cos(θ) = = hyp 5

cos(8)

opp 4 tan(θ) = = adj 3

tan(0)

*

*

*

239

*

—

adj

—

3

--

—

—

4

--

*

*

*

*

*

Exercises sin(✓) For #1-7, use the definition of tangent, that tan(✓) = cos(✓) , to identify the given value. You can use the chart on page 227 for help.

1.) tan

⇡ 3

5.) tan

⇡ 6

2.) tan

⇡ 4

6.) tan

⇡ 4

3.) tan

⇡ 6

7.) tan

⇡ 3

4.) tan(0)

Suppose that is a real number, that 0   ⇡2 , and that cos( ) = 13 . Use Lemmas 7-12 from the previous chapter, the definition of tangent, and the periods of sine, cosine, and tangent to find the following values.

8.) sin( )

14.) cos(

)

9.) tan( )

15.) sin(

)

10.) sin( + ⇡2 )

16.) cos( + 2⇡)

⇡ 2)

17.) sin( + 2⇡)

11.) cos(

12.) cos( + ⇡)

18.) tan( + ⇡)

13.) sin( + ⇡) 240

Match the numbered piecewise defined functions with their lettered graphs below. ( sin(x) 19.) f (x) = tan(x)

if x ∈ [0, ∞); and if x ∈ (− π2 , 0).

( tan(x) 20.) g(x) = cos(x)

if x ∈ (0, π2 ); and if x ∈ (−∞, 0].

( tan(x) 21.) h(x) = cos(x)

if x ∈ [0, π2 ); and if x ∈ (−∞, 0).

( sin(x) 22.) p(x) = tan(x)

if x ∈ (0, ∞); and if x ∈ (− π2 , 0].

A.)

/2

B.)

C.)

241

Exercises Exercises Exercises Exercises cos(6), and Exercises tan(6) for the angles Exercises

Find sin(6), 6 given below. You might might Find Find sin(6), cos(6), and tan(6) for the 6 given You theorem tobelow. findbelow. the length a side have begin by using the Pythagorean might sin(6), cos(6), and tan(6) forforangles the 6 6given You might Findto sin(6), cos(6), and tan(6) theangles angles given below. Youof Use Propostion 13 to find sin(θ), cos(θ), and tan(θ) for the angles θ given Find cos(6), and fortheorem thetheorem angles 6toto given below. to 6find the length of You a side haveFind tohave begin by using the Pythagorean that issin(6), not labelled with length. aPythagorean find the of a aside have toto begin by the theorem find thelength length ofmight side begin byusing using thetan(6) Pythagorean might sin(6), cos(6), and tan(6) forbythe angles given below. You below. You might have to begin using the Pythagorean Theorem to find theorem to find the length of a side have to begin by using the Pythagorean that is not labelled with length. a that with a alength. thatisbegin isnot notlabelled labelled withPythagorean length. find the length of a side have by the thetolength a using side with that is length. not labelledtheorem with a to length. that is1.) not of labelled a 4.) that is not labelled with a length. 1.) 1.)1.) 4.) 4.)4.) 23.) 26.) 1.) 4.) 1.) 4.)

12 12 1212 12 12

2.) 2.) 2.)2.) 24.) 2.) 2.)

8 8 88 88

10 10 1010 10 10

5.) 5.)

5.) 5.)5.) 27.) 5.)

5 5 55 55

17 17 1717 1717 15 15

8 8 88 88

15 1515 15

3.) 3.) 3.)3.) 25.) 3.) 3.)

6.)

7 7 77 77 25 25 25 25 25 25

6.)

6.) 6.)6.) 28.) 6.)

3 3 33 33 218 218 218

218218 218

242

2 2 22 22