Answer to Question #341923 in Trigonometry for Burog

Question

1.If sin = 12, and is in QI, find sec data .

2.Given cos =35 , is in QIV, find csc data.

3.Find the exact values of the circular functions whose =-3π/4.

Answer 1

1. As it is in Q1, then all functions are positive.

$sin \alpha =1/2$

$cos^2\alpha=1-sin^2\alpha=1-1/4=3/4$

$cos \alpha =\sqrt {3/4}=\sqrt{3}/2$

$sec\alpha=1/cos \alpha=1/(\sqrt{3}/2)=2/\sqrt3$

2. As it is in QIV, then sin<0 and csc<0

functions are positive.

$cos^2\alpha=3/5$

$sin^2\alpha=1-cos^2\alpha=1-9/25=16/25$

$sin \alpha=\sqrt{16/25}=-4/5$

$csc=1/sin\alpha=1/(-4/5)=-5/4$

3. $3\pi/4=135°=3x45°$

Of course 30/60/90 and 45/45/90 are the only two triangles students are expected to know "exactly." We have to know the multiples of these triangles in the other quadrant. Here we have 45/45/90 in the second quadrant.

We start from cos (45°)=sin (45°)=$1/\sqrt2$

The angle

135∘

is supplementary to

45∘

, so has the opposite cosine and the same sine.

$cos({3\pi}/4)= - cos(\pi - {3\pi}/4)=- cos(\pi/4)= - 1/\sqrt{2}$

$sin({3\pi}/4)=sin( \pi - {3\pi}/4) = sin(\pi/4) = 1/\sqrt{2}$

$tan({3\pi}/4) = {sin({3\pi}/4)}/{cos({3\pi}/4)} = {(1/\sqrt{2})}/{(-1/\sqrt{2})}=-1$

$sec({3\pi}/4)=1/cos({3\pi}/4) = - \sqrt{2}$

$csc({3\pi}/4)=1/sin({3\pi}/4) = \sqrt{2}$

$cot({3\pi}/4)=1/tan({3\pi}/4) = -1$