Using the pythagorean identity, find the value of sin? and tan? if cos?= -1/2

Question

Using the pythagorean identity, find the value of sin⊗ and tan⊗ if cos⊗= -1/2

Answer 1

By definition, in a right angled triangle with a angle being θ, cosθ= adjacent side/hypotenuse

and sinθ = distant side/ hypotenuse and also, tanθ=distant side/ adjacent side

In the given problem, cosθ=-1/2,

hence, in the given right angled triangle with a angle being θ the ratio, adjacent side/hypotenuse=-1/2

Applying Pythagorean hypothesis for (Hypotenuse)²= (Base)2+(Perpendicular)2

                                                                          = (adjacent side)2 + (distant side)²

Thus, switching sides,

[(distant side)2+[(adjacent side)2=(Hypotenuse)2

=>[(distant side)2=(Hypotenuse)2-(adjacent side)2 ............subtracting(adjacent side)2 from both sides

=>(distant side)2/(Hypotenuse)2=1-[(adjacent side)2/(Hypotenuse)2], .......dividing both sides by (Hypotenuse)2

=>(distant side/Hypotenuse)2=1-(adjacent side/Hypotenuse)2............adjusting operators

=> (distant side/Hypotenuse)2=1-(-1/2)2.................substituting given values

=> (distant side/Hypotenuse)2=1-(1/4) ............adjusting operators

=> (distant side/Hypotenuse)2=3/4............adjusting operators

=> (distant side/Hypotenuse)=±√3/2

=>sinθ=±√3/2.... By definition

Since cosθ is negative, both positive and negative values are true as θ is either >90 degrees and < 180 degrees, or > 180 degrees and <270 degrees, thus sinθ=√3/2 or -√3/2

and

for  θ > 90 degrees and < 180 degrees, tanθ= distant side/ adjacent side

                                                                =(distant side/Hypotenuse)/(adjacent side/hypotenuse)

                                                                =(√3/2)/(-1/2)

                                                                =-√3

for θ > 180 degrees and <270 degrees, tanθ= distant side/ adjacent side
                                                                =(distant side/Hypotenuse)/(adjacent side/hypotenuse)
                                                                =(-√3/2)/(-1/2)
                                                                =√3

Answer 2

SOHCAHTOA is one of the main things to keep in mind in this question because it will help set up the equation for you.

1. Equations

Sin= opp/hyp

Cos= adj/hyp

Tan=opp/adj

a²+b²=c²

 

2. Cos = -1/2 therefore -1 = adj side and 2 = hypotenuse

1² + b² = 2²

1 + b² = 4

-1          -1

 

b²= 3

take the square root of both sides

b= √3

opp = √3

 

3. Going back to SOH CAH TOA

Sin = √3/2

Tan = -√3/1