Given sin=35andcos=45 find tan.

Kindly allow us to rewrite the question:

(i)"Given sin35 and cos45, find tan35 and tan45".

Or

(ii)Given sin A =3/5 and cosA =4/5 find tan A.

Let us start with the basics which explains why we corrrected as above:

Definition: Consider a right angled triangle ABC with right angle at B. Then,

sineA = opposite side/hypotenuse=BC/AC,

cosine A=adjascent side/hypotenuse=AB/AC and

tangent A = opoosite side/adjascent side=BC/AB.

Note: the sineA can take values 0  to 1  when A takes values from 0 to 90 deg , sinA from  1 to 0 for A =90 deg to 180 deg, sinA from 0 to -1 for A=180 to 270 deg and  -1 to 0 for A=270 to 360 degree. Similarly the cosine and tangent ratios of an angle take values are as below:

A           =    0 to    90, 90 to   180 , 180 to  270, 270 to 360

cosA      =    1 to     0 ,   0 to   -1,   -1   to     0,   0   to     1

Tangent A= 0+infinite,  -infinte to 0,   0 to infinite,  -infinite to 0.

From the above we see that -1<=sin of an angle<=+1 and -1<=cos of an angle<=+1 and -infinite <=tan of an angle<= infinite. The sine and cosine curves are continuous, where as the graph of tangent of an angle is discontinous at 90 deg and 270 degree with infinite gap.

Therefore, the sine of an angle  cannot be 35 . Similarly, the   cosine of an angle cannot be 45. Therefore, the given question  to be meaningful should read  like :

Given sin35  and cos45,  find,   tan 35 and tan 45.

and we proceed for on the method to reach the solution:

We, know by trigonometry: (sin X)^2+(cos X)^= 1, is an identity and since (3/5)^2+(4/5)^2=1, We can say: "Given sinA=3/5 and cosA =4/5, find tan A". This is solved in second part below:

(i)

Therefore, tanx = sinX/cosX= sinX/sqrt[1-(sin X)^2]strong> or

tanX=sqrt[1-(cosX)^2]/cosX.

Let sin35 = x,

Tangent 35 = sin35/cos35 =sin35/sqrt[1-(sin35)^2]= x/(1-x^2). = 0.573576436sqrt/(1-0.573576436^2) =0.700207538

Given cos45=y,say.

Then, tangent 45 = sin45/cos45

=y/y=1.

As y=sin45 =cos45 = 1/sqrt2

(ii)

Given sinA =3/5 and cosA =4/5 , find tanA.

By trigonometry,

tanA = sinA/cosA. Substitute the given values of sinA and cosA to get the value of tan A.

tan A = (3/5)/(4/5) =3*5/(4*5) =3/4 =0.75

Hope this helps.