Solve on the interval $\left[0^{\circ}, 360^{\circ}\right)$. Hint: Use an Identity!

$2 \sec ^2 x+\tan ^2 x=3$

Use the identity sec 2 x = 1 + tan 2 x to rewrite the equation in terms of tan 2 x .
Solve for tan 2 x to get tan 2 x = 3 1 ​ .
Take the square root of both sides to get tan x = ± 3 ​ 1 ​ .
Find the angles x in the interval [ 0 ∘ , 36 0 ∘ ) such that tan x = 3 ​ 1 ​ and tan x = − 3 ​ 1 ​ , which are 3 0 ∘ , 15 0 ∘ , 21 0 ∘ , 33 0 ∘ . The final answer is { 3 0 ∘ , 15 0 ∘ , 21 0 ∘ , 33 0 ∘ } ​ .

Explanation

Understanding the Problem We are asked to solve the trigonometric equation 2 sec 2 x + tan 2 x = 3 on the interval [ 0 ∘ , 36 0 ∘ ) . We will use trigonometric identities to simplify the equation and find the solutions within the given interval.

Using Trigonometric Identity We know the trigonometric identity sec 2 x = 1 + tan 2 x . We can use this identity to rewrite the given equation in terms of tan 2 x .

Substitution Substitute sec 2 x = 1 + tan 2 x into the equation 2 sec 2 x + tan 2 x = 3 :
2 ( 1 + tan 2 x ) + tan 2 x = 3

Simplifying the Equation Simplify the equation: 2 + 2 tan 2 x + tan 2 x = 3 3 tan 2 x = 1

Solving for tan 2 x Solve for tan 2 x :
tan 2 x = 3 1 ​

Taking the Square Root Take the square root of both sides: tan x = ± 3 ​ 1 ​

Finding the Angles Find the angles x in the interval [ 0 ∘ , 36 0 ∘ ) such that tan x = 3 ​ 1 ​ and tan x = − 3 ​ 1 ​ .
For tan x = 3 ​ 1 ​ , the reference angle is 3 0 ∘ . Since tangent is positive in the first and third quadrants, the solutions are x = 3 0 ∘ and x = 18 0 ∘ + 3 0 ∘ = 21 0 ∘ .
For tan x = − 3 ​ 1 ​ , the reference angle is 3 0 ∘ . Since tangent is negative in the second and fourth quadrants, the solutions are x = 18 0 ∘ − 3 0 ∘ = 15 0 ∘ and x = 36 0 ∘ − 3 0 ∘ = 33 0 ∘ .

Final Answer Therefore, the solutions are x = 3 0 ∘ , 15 0 ∘ , 21 0 ∘ , 33 0 ∘ .


Examples
Imagine you're designing a solar panel array, and you need to optimize the angle of the panels to maximize sunlight absorption throughout the day. Trigonometric equations like the one we solved can help you determine the optimal angles at different times of the year, ensuring that your solar panels are always positioned to capture the most energy. This blend of math and engineering allows for efficient and sustainable energy solutions.

Answered by GinnyAnswer | 2025-07-03

To solve the equation 2 sec 2 x + tan 2 x = 3 , we used the identity sec 2 x = 1 + tan 2 x to rewrite and simplify the equation to find tan 2 x = 3 1 ​ . This led to the solutions x = 3 0 ∘ , 15 0 ∘ , 21 0 ∘ , 33 0 ∘ within the interval [ 0 ∘ , 36 0 ∘ ) .
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Answered by Anonymous | 2025-07-04