Homework Help

If 2sin^2θ + 5sinθ  - 3 = 0. Find the value of tan^4θ + tan^2θ +9

user profile pic

nasirjam | Student, Grade 9 | (Level 2) Honors

Posted April 5, 2013 at 10:31 PM via web

dislike 0 like

If 2sin^2θ + 5sinθ  - 3 = 0. Find the value of tan^4θ + tan^2θ +9

1 Answer | Add Yours

user profile pic

violy | High School Teacher | (Level 3) Assistant Educator

Posted April 6, 2013 at 1:06 AM (Answer #1)

dislike 1 like

Let set `u = sintheta`

So we will have: 

`2u^(2) +5u - 3 = 0`

Factor the left side. 

`(2u - 1)(u + 3) = 0`

Equate each factor to zero. 

`2u - 1 = 0` ;`u + 3 = 0`

Add 1 on both sides, to isolate the 2u on left side. 

`2u = 1`

Divide both sides by 2 to isolate the u on left side. 

` ` `u = 1/2`

Plug-in u = sinx. 

`sintheta = 1/2`

Therefore, theta = pi/6, 5pi/6. 

Note that for u + 3 = 0, we will have u = -3. Which will  leads to 

sintheta = -3. Which has no solution, since range of sinx is from -1 to 1. 

Now, we plug-in the theta we got.

`tan^(4)(pi/6) + tan^(2)(pi/6) + 9 = (sqrt(3)/3)^4 + (sqrt(3)/3)^2 + 9`

`9/81 + 3/9 + 9 = 9/81 + 27/81 + 81/9 = 85/9`

Therefore, the value for theta = pi/6 is 13. 

Plug-in theta = 5pi/6.

`tan^(4)((5pi)/6) + tan^(2)((5pi)/6) + 9 = (-sqrt(3)/3)^4 + (-sqrt(3)/3)^2 + 9 = 85/9`

Therefore the value is 85/9.

Join to answer this question

Join a community of thousands of dedicated teachers and students.

Join eNotes