<< Chapter < Page Chapter >> Page >

Given that sin α = 4 5 and α lies in quadrant IV, find the exact value of cos ( α 2 ) .

2 5

Got questions? Get instant answers now!

Finding the measurement of a half angle

Now, we will return to the problem posed at the beginning of the section. A bicycle ramp is constructed for high-level competition with an angle of θ formed by the ramp and the ground. Another ramp is to be constructed half as steep for novice competition. If tan θ = 5 3 for higher-level competition, what is the measurement of the angle for novice competition?

Since the angle for novice competition measures half the steepness of the angle for the high level competition, and tan θ = 5 3 for high competition, we can find cos θ from the right triangle and the Pythagorean theorem so that we can use the half-angle identities. See [link] .

3 2 + 5 2 = 34 c = 34
Image of a right triangle with sides 3, 5, and rad34. Rad 34 is the hypotenuse, and 3 is the base. The angle formed by the hypotenuse and base is theta. The angle between the side of length 3 and side of length 5 is a right angle.

We see that cos θ = 3 34 = 3 34 34 . We can use the half-angle formula for tangent: tan θ 2 = 1 cos θ 1 + cos θ . Since tan θ is in the first quadrant, so is tan θ 2 .

tan θ 2 = 1 3 34 34 1 + 3 34 34 = 34 3 34 34 34 + 3 34 34 = 34 3 34 34 + 3 34 0.57

We can take the inverse tangent to find the angle: tan 1 ( 0.57 ) 29.7° . So the angle of the ramp for novice competition is 29.7° .

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

Access these online resources for additional instruction and practice with double-angle, half-angle, and reduction formulas.

Key equations

Double-angle formulas sin ( 2 θ ) = 2 sin θ cos θ cos ( 2 θ ) = cos 2 θ sin 2 θ = 1 2 sin 2 θ = 2 cos 2 θ 1 tan ( 2 θ ) = 2 tan θ 1 tan 2 θ
Reduction formulas sin 2 θ = 1 cos ( 2 θ ) 2 cos 2 θ = 1 + cos ( 2 θ ) 2 tan 2 θ = 1 cos ( 2 θ ) 1 + cos ( 2 θ )
Half-angle formulas sin α 2 = ± 1 cos α 2 cos α 2 = ± 1 + cos α 2 tan α 2 = ± 1 cos α 1 + cos α = sin α 1 + cos α = 1 cos α sin α

Key concepts

  • Double-angle identities are derived from the sum formulas of the fundamental trigonometric functions: sine, cosine, and tangent. See [link] , [link] , [link] , and [link] .
  • Reduction formulas are especially useful in calculus, as they allow us to reduce the power of the trigonometric term. See [link] and [link] .
  • Half-angle formulas allow us to find the value of trigonometric functions involving half-angles, whether the original angle is known or not. See [link] , [link] , and [link] .

Section exercises

Verbal

Explain how to determine the reduction identities from the double-angle identity cos ( 2 x ) = cos 2 x sin 2 x .

Use the Pythagorean identities and isolate the squared term.

Got questions? Get instant answers now!

Explain how to determine the double-angle formula for tan ( 2 x ) using the double-angle formulas for cos ( 2 x ) and sin ( 2 x ) .

Got questions? Get instant answers now!

We can determine the half-angle formula for tan ( x 2 ) = 1 cos x 1 + cos x by dividing the formula for sin ( x 2 ) by cos ( x 2 ) . Explain how to determine two formulas for tan ( x 2 ) that do not involve any square roots.

1 cos x sin x , sin x 1 + cos x , multiplying the top and bottom by 1 cos x and 1 + cos x , respectively.

Got questions? Get instant answers now!

For the half-angle formula given in the previous exercise for tan ( x 2 ) , explain why dividing by 0 is not a concern. (Hint: examine the values of cos x necessary for the denominator to be 0.)

Got questions? Get instant answers now!

Algebraic

For the following exercises, find the exact values of a) sin ( 2 x ) , b) cos ( 2 x ) , and c) tan ( 2 x ) without solving for x .

If sin x = 1 8 , and x is in quadrant I.

a) 3 7 32 b) 31 32 c) 3 7 31

Got questions? Get instant answers now!

If cos x = 2 3 , and x is in quadrant I.

Got questions? Get instant answers now!

If cos x = 1 2 , and x is in quadrant III.

a) 3 2 b) 1 2 c) 3

Got questions? Get instant answers now!

Questions & Answers

Cos45/sec30+cosec30=
dinesh Reply
Cos 45 = 1/ √ 2 sec 30 = 2/√3 cosec 30 = 2. =1/√2 / 2/√3+2 =1/√2/2+2√3/√3 =1/√2*√3/2+2√3 =√3/√2(2+2√3) =√3/2√2+2√6 --------- (1) =√3 (2√6-2√2)/((2√6)+2√2))(2√6-2√2) =2√3(√6-√2)/(2√6)²-(2√2)² =2√3(√6-√2)/24-8 =2√3(√6-√2)/16 =√18-√16/8 =3√2-√6/8 ----------(2)
exercise 1.2 solution b....isnt it lacking
Miiro Reply
I dnt get dis work well
john Reply
what is one-to-one function
Iwori Reply
what is the procedure in solving quadratic equetion at least 6?
Qhadz Reply
Almighty formula or by factorization...or by graphical analysis
Damian
I need to learn this trigonometry from A level.. can anyone help here?
wisdom Reply
yes am hia
Miiro
tanh2x =2tanhx/1+tanh^2x
Gautam Reply
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)=cotb ... pls some one should help me with this..thanks in anticipation
favour Reply
f(x)=x/x+2 given g(x)=1+2x/1-x show that gf(x)=1+2x/3
Ken Reply
proof
AUSTINE
sebd me some questions about anything ill solve for yall
Manifoldee Reply
cos(a+b)+cos(a-b)/sin(a+b)-sin(a-b)= cotb
favour
how to solve x²=2x+8 factorization?
Kristof Reply
x=2x+8 x-2x=2x+8-2x x-2x=8 -x=8 -x/-1=8/-1 x=-8 prove: if x=-8 -8=2(-8)+8 -8=-16+8 -8=-8 (PROVEN)
Manifoldee
x=2x+8
Manifoldee
×=2x-8 minus both sides by 2x
Manifoldee
so, x-2x=2x+8-2x
Manifoldee
then cancel out 2x and -2x, cuz 2x-2x is obviously zero
Manifoldee
so it would be like this: x-2x=8
Manifoldee
then we all know that beside the variable is a number (1): (1)x-2x=8
Manifoldee
so we will going to minus that 1-2=-1
Manifoldee
so it would be -x=8
Manifoldee
so next step is to cancel out negative number beside x so we get positive x
Manifoldee
so by doing it you need to divide both side by -1 so it would be like this: (-1x/-1)=(8/-1)
Manifoldee
so -1/-1=1
Manifoldee
so x=-8
Manifoldee
SO THE ANSWER IS X=-8
Manifoldee
so we should prove it
Manifoldee
x=2x+8 x-2x=8 -x=8 x=-8 by mantu from India
mantu
lol i just saw its x²
Manifoldee
x²=2x-8 x²-2x=8 -x²=8 x²=-8 square root(x²)=square root(-8) x=sq. root(-8)
Manifoldee
I mean x²=2x+8 by factorization method
Kristof
I think x=-2 or x=4
Kristof
x= 2x+8 ×=8-2x - 2x + x = 8 - x = 8 both sides divided - 1 -×/-1 = 8/-1 × = - 8 //// from somalia
Mohamed
i am in
Cliff
1KI POWER 1/3 PLEASE SOLUTIONS
Prashant Reply
hii
Amit
how are you
Dorbor
well
Biswajit
can u tell me concepts
Gaurav
Find the possible value of 8.5 using moivre's theorem
Reuben Reply
which of these functions is not uniformly cintinuous on (0, 1)? sinx
Pooja Reply
helo
Akash
hlo
Akash
Hello
Hudheifa
which of these functions is not uniformly continuous on 0,1
Basant Reply
Practice Key Terms 3

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