true false on trig identities.

Question

a) If tan(θ) = 5 /7 , then sin(θ) = 5 and cos(θ) = 7.

For this one I said the statement was true because tan=sin/cos. but my friend suggested false because its sin of an angle =5 which means that 5 is equal to y/hypotenuse. So now I don't know what to think about this.

(b) sin(α^2) + cos(α^2) = 1.

For this one I said it was true because that's the identity, but then my friend suggested false because the angle is being squared not the whole term. I don't really see the difference but now I don't know.

(c) cot^−1(x) = 1/ cot(x)

This one I don't know what to think. cot=1/tan, so the inverse is tan. 1/cot=1/1/tan=tan, but this answer is also the reciprocal, so I don't know if this statement is true.

Answer 1

1) tan θ = sin Θ / cos θ

 

    tan Θ = 5 / 7  only means that the ratio of sinΘ

                          to cosΘ is 5 /7 ;  there are countless

                          numbers that could be in this ratio;

                          Remember:  -1 ≤ sinX ≤ 1

                           and             -1 ≤ cosX ≤ 1

 

3)  to refer to the inverse of  cos X 

 

    We would write: [cos (X)]^-1 = 1 / cos X

 

    We also write:

 

    cos ^-1 (X)  = arccos X

Answer 2

Hi Olivia

 

a)  The statement is false.  I would say that opposite = 5 and adjacent = 7.  You could then use Pythagorus to find the hypothenuse and find sin and cos.  Also recall that -1≤sin≤1 and -1≤cos≤1

 

b) I agree with your friend; this statement is false.  The identity squares the sin and cos, not the angle.  In the statement α the angle is being squared not the sin and cos of α.

 

c)  This is also false.  cot^-1(x) is not 1/cot(x) but rather it's the inverse cot(x).  In English, what angle has a co-tan of x.  In the case of inverse trig functions, the x is not the angle.

 

Hope this helps.