Mid-Chapter Quiz
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Use the fact that cot θ = 1tan θ and rewrite 75^(∘) as 120^(∘)-45^(∘). Then, use the Difference of Angles Identity for a tangent function.
2-sqrt(3)
The cotangent of an angle is defined as the quotient of 1 and the tangent of the angle. Using this fact, we can rewrite our expression. cot 75^(∘) = 1/tan 75^(∘) Let's recall the values of the three main trigonometric functions for the most important angles.
| Trigonometric Values for Special Angles | ||
|---|---|---|
| Sine | Cosine | Tangent |
| sin 0^(∘)=0 | cos 0^(∘)=1 | tan 0^(∘)=0 |
| sin 30^(∘)=1/2 | cos 30^(∘)=sqrt(3)/2 | tan 30^(∘)=sqrt(3)/3 |
| sin 45^(∘)=sqrt(2)/2 | cos 45^(∘)=sqrt(2)/2 | tan 45^(∘)=1 |
| sin 60^(∘)=sqrt(3)/2 | cos 60^(∘)=1/2 | tan 60^(∘)=sqrt(3) |
| sin 90^(∘) = 1 | cos 90^(∘) = 0 | - |
| sin 120^(∘)= sqrt(3)/2 | cos 120^(∘)= - 1/2 | tan 120^(∘)= - sqrt(3) |
| sin 135^(∘)= sqrt(2)/2 | cos 135^(∘)= - sqrt(2)/2 | tan 135^(∘)= - 1 |
| sin 150^(∘)= 1/2 | cos 150^(∘)= - sqrt(3)/2 | tan 150^(∘)= - sqrt(3)/3 |
| sin 180^(∘)= 0 | cos 180^(∘)= - 1 | tan 180^(∘)= 0 |
| sin 210^(∘)= - 1/2 | cos 210^(∘)= - sqrt(3)/2 | tan 210^(∘)= sqrt(3)/3 |
| sin 240^(∘)= - sqrt(3)/2 | cos 240^(∘)= - 1/2 | tan 240^(∘)= sqrt(3) |
| sin 270^(∘)= - 1 | cos 270^(∘)= 0 | - |
| sin 300^(∘)= - sqrt(3)/2 | cos 300^(∘)= 1/2 | tan 300^(∘)= - sqrt(3) |
| sin 330^(∘) = - 1/2 | cos 330^(∘) = sqrt(3)/2 | tan 330^(∘) = - sqrt(3)/3 |
| sin 360^(∘)= 0 | cos 360^(∘)= 1 | tan 360^(∘)= 0 |
tan 120^(∘)= - sqrt(3), tan 45^(∘)= 1
We found that the value of tan (120^(∘)-45^(∘)), and therefore the value of tan 75^(∘), is - sqrt(3)-11-sqrt(3). Let's now find its reciprocal. To do so, we switch the numerator and the denominator. ccc Expression & & Reciprocal [0.8em] - sqrt(3)-1/1-sqrt(3) & & 1-sqrt(3)/- sqrt(3)-1 Finally, let's simplify the reciprocal to obtain the value of 1tan 75^(∘). We will need to rationalize the denominator of the expression. Remember, 1tan 75^(∘) is the same as the value of cot 75^(∘) (the original expression).
Identity Property of Multiplication
Rewrite 1 as sqrt(3)-1/sqrt(3)-1
Multiply fractions
Distribute (sqrt(3)-1)
Distribute 1 & - sqrt(3) & - 1
( sqrt(a) )^2 = a
Add and subtract terms
Put minus sign in front of fraction
Write as a difference of fractions
Cancel out common factors
Calculate quotient
Distribute (- 1)
Commutative Property of Addition
| Property Used | Equivalency Found |
|---|---|
| Definition of Cotangent | cot 75^(∘) = 1/tan 75^(∘) |
| Substitution Property of Equality | 1/tan 75^(∘) = 1/tan ( 120^(∘)- 45^(∘)) |
| Difference of Angles Identity | tan (120^(∘)- 45^(∘)) = tan 120^(∘)-tan 45^(∘)/1+tan 120^(∘) tan 45^(∘) |
| Values of Tangent | tan 120^(∘)-tan 45^(∘)/1+tan 120^(∘) tan 45^(∘) = - sqrt(3)-1/1-sqrt(3) |
| Reciprocal Fractions | 1/tan ( 120^(∘)- 45^(∘)) = 1- sqrt(3)/- sqrt(3)-1 |
| Simplification | 1- sqrt(3)/- sqrt(3)-1 = 2-sqrt(3) |
| Transitive Property | cot 75^(∘) = 2-sqrt(3) |