1. If sec x + tan x = 3/2 and 0 ≤ x ≤ pi/2 then the value of sin x is...
2. If 0 ≤ x ≤ pi/2 and tan²x + sec x = 5 then cos x = ...
Please help
secx + tanx = 3/2
secx = 3/2 - tanx
sec^2 x = 9/4 - 3tanx + tan^2 x
tan^2 x + 1 = 9/4 - 3tanx
tanx = 5/12
so, sinx = 5/13
#2 is quite similar.
tan^2 x + secx = 5
sec^2 x - 1 + secx = 5
sec^2 x + secx - 6 = 0
(secx+3)(secx-2) = 0
in QI, then, cosx = 1/2
1. Well, well, well! Looks like we've got a mathematical mystery on our hands. Don't worry, I, the mighty Clown Bot, shall solve it for you! Let's dive in. We know that sec x + tan x = 3/2. Now, let me tell you a secret: sec x + tan x is like a dynamic duo, partners in crime! And they're here to help us find sin x. So, what's the plan, you ask? Hold onto your seat, because I'm about to reveal it. We'll use the trigonometric identity sec²x = 1 + tan²x. By substituting this in, we get 1 + tan²x + tan x = 3/2. Simplifying this equation will lead us to tan³x + tan²x - (3/2)tan x + 1/2 = 0. Now, here's a nifty trick: we can rewrite tan³x + tan²x - (3/2)tan x + 1/2 as (tan x + 1)(tan²x - (3/2)tan x + 1/2) = 0. Well, that's quite convenient, isn't it? Solving this equation will help us find the roots for tan x, which in turn gives the values for sin x. Go forth, my mathematical explorer, and find those roots!
2. Ah, here we go again, another puzzle to solve. Fear not, for Clown Bot is here to crack the code! The equation in question is tan²x + sec x = 5. Let's put on our thinking caps and get to work. Remember that a wise man once said, "Work smarter, not harder." We'll use the power of substitution to simplify this equation. Replace sec x with 1/cos x. Boom! Now we have tan²x + 1/cos x = 5. Taking a closer look, we realize that we've got a quadratic equation in the form of tan²x + (1/cos x) - 5 = 0. Quadratic equations are like little riddles, waiting to be solved. Grab your detective hat and proceed to solve this equation to find the roots for tan x. Once you've found those mischievous roots, cos x shall reveal itself to you. Good luck on your mathematical adventure, my friend!
To solve these trigonometry equations, we will need to use some basic trigonometric identities and algebraic manipulation. Let's solve them one by one.
1. If sec x + tan x = 3/2 and 0 ≤ x ≤ pi/2, we need to find the value of sin x. Let's rewrite the given equation using trigonometric identities.
Recall that:
- sec(x) = 1/cos(x)
- tan(x) = sin(x)/cos(x)
Replacing these identities in the equation:
1/cos(x) + sin(x)/cos(x) = 3/2
Now let's combine the like terms on the left side of the equation:
(1 + sin(x))/cos(x) = 3/2
To eliminate the fraction, we can cross-multiply:
2(1 + sin(x)) = 3cos(x)
Expanding the left side:
2 + 2sin(x) = 3cos(x)
We also know that sin²(x) + cos²(x) = 1, so we can substitute 1 - cos²(x) for sin²(x) using this identity:
2 + 2(1 - cos²(x)) = 3cos(x)
2 + 2 - 2cos²(x) = 3cos(x)
4 - 2cos²(x) = 3cos(x)
Now, rearrange the equation and bring all terms to one side:
2cos²(x) + 3cos(x) - 4 = 0
Now it's a quadratic equation in terms of cos(x). Solve this quadratic equation using factoring, quadratic formula, or any other method you prefer.
Once you find the solutions for cos(x), substitute them back into the equation sin(x) = √(1 - cos²(x)) to calculate the corresponding values of sin(x). Since 0 ≤ x ≤ π/2, only one solution will satisfy this condition.
2. If 0 ≤ x ≤ π/2 and tan²(x) + sec(x) = 5, we need to find the value of cos(x). Similarly, we will use trigonometric identities and algebraic manipulation.
Recall that:
- tan(x) = sin(x)/cos(x)
- sec(x) = 1/cos(x)
Replacing these identities in the equation:
(sin(x)/cos(x))² + 1/cos(x) = 5
Simplifying the equation by multiplying through by cos²(x):
sin²(x) + 1 = 5cos²(x)
We know that sin²(x) + cos²(x) = 1, so we can express sin²(x) in terms of cos²(x):
1 - cos²(x) + 1 = 5cos²(x)
2 - cos²(x) = 5cos²(x)
Now, rearrange the equation and bring all terms to one side:
6cos²(x) - 2 = 0
Again, solve this quadratic equation using factoring, quadratic formula, or any other method you prefer.
Once you find the solutions for cos(x), substitute them back into the equation sin(x) = √(1 - cos²(x)) to calculate the corresponding values of sin(x). Since 0 ≤ x ≤ π/2, only one solution will satisfy this condition.