Calculate value of E(a)=(4/5)tg(a)+(5/12)sin(2a), if cos(a) = -4/5, a ∈ (-pi; -pi/2)

[Christofor]

Calculate value of E(a)=(4/5)tg(a)+(5/12)sin(2a), if cos(a) = -4/5, a ∈ (-pi; -pi/2)

Hi, I am a student and I do exercises at home from past exams to get ready for the test for next summer and I have a problem that I can't finish it.
Calculate the value of the expression E(a)=(4/5)tg(a)+(5/12)sin(2a), if it is known that cos(a) and a ∈ (-pi; -pi/2). (see the photo)

What I did: (4/5)tg(a)+(5/12)sin(2a) = (4/5)tg(a)+(5/12)*2sin(a)cos(a)
(4/5)tg(a)+(5/6)sin(a)cos(a) = (4/5)tg(a)+(5/6)sin(a)*(-4/5)
4/5(tg(a)-(5/6)sin(a)) and here I stuck.


Here are given points for: 1p for obtaining sin(a)^2 = 9/25 ( here is sin(a)*sin(a) not sin(a^2) )
1p for obtaining sin(a) = -3/5
1p for obtaining sin(2a) = 24/25
1p for obtaining tg(a) =3/4
1p for obtaining E(a) =1

Can someone help me, please?

 

[Dr.Peterson]

Hi, I am a student and I do exercises at home from past exams to get ready for the test for next summer and I have a problem that I can't finish it.
Calculate the value of the expression E(a)=(4/5)tg(a)+(5/12)sin(2a), if it is known that cos(a)=-4/5 and a ∈ (-pi; -pi/2). (see the photo)

What I did: (4/5)tg(a)+(5/12)sin(2a) = (4/5)tg(a)+(5/12)*2sin(a)cos(a)
(4/5)tg(a)+(5/6)sin(a)cos(a) = (4/5)tg(a)+(5/6)sin(a)*(-4/5)
4/5(tg(a)-(5/6)sin(a)) and here I stuck.


Here are given points for: 1p for obtaining sin(a)^2 = 9/25 ( here is sin(a)*sin(a) not sin(a^2) )
1p for obtaining sin(a) = -3/5
1p for obtaining sin(2a) = 24/25
1p for obtaining tg(a) =3/4
1p for obtaining E(a) =1

Can someone help me, please?

I would start by finding what sin(a) and tg(a) are, from the given information.

I tend to do these in terms of the definition of the trig functions from a point (x,y) on the terminal ray in standard position. Since cos(a) = x/r, we can take x = -4 and r = 4. From that, find y = ±sqrt(r^2 - x^2), choose the sign based on the given quadrant, and use these to find the sine and tangent. Then just plug everything in.

Note that this is the order they expect, first finding sin(a). There are several ways to do that, such as using identities and knowledge of the signs of trig functions in each quadrant; but mine feels cleanest.

 

[Christofor]

I would start by finding what sin(a) and tg(a) are, from the given information.

I tend to do these in terms of the definition of the trig functions from a point (x,y) on the terminal ray in standard position. Since cos(a) = x/r, we can take x = -4 and r = 4. From that, find y = ±sqrt(r^2 - x^2), choose the sign based on the given quadrant, and use these to find the sine and tangent. Then just plug everything in.

Note that this is the order they expect, first finding sin(a). There are several ways to do that, such as using identities and knowledge of the signs of trig functions in each quadrant; but mine feels cleanest.

Thanks, I appreciate.