Let's solve the equation step by step to understand the given expression:
[tex]\[
\tan^2 \theta + 74 \sin^2 \beta = 34
\][/tex]
First, let's rewrite the equation clearly:
[tex]\[
\tan^2(\theta) + 74 \sin^2(\beta) = 34
\][/tex]
### Step-by-Step Solution
1. Identify the given equation:
The equation provided relates the trigonometric functions [tex]\(\tan\)[/tex] (tangent) and [tex]\(\sin\)[/tex] (sine) for angles [tex]\(\theta\)[/tex] and [tex]\(\beta\)[/tex] respectively.
2. Understand the trigonometric functions:
- [tex]\(\tan(\theta)\)[/tex] is the tangent of angle [tex]\(\theta\)[/tex], defined as [tex]\(\tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)}\)[/tex].
- [tex]\(\sin(\beta)\)[/tex] is the sine of angle [tex]\(\beta\)[/tex], defined as the ratio of the length of the opposite side to the hypotenuse in a right-angled triangle.
3. Square the functions:
- The squared tangent function: [tex]\(\tan^2(\theta)\)[/tex].
- The squared sine function multiplied by 74: [tex]\(74 \sin^2(\beta)\)[/tex].
4. Combine the expressions:
Sum the squared tangent of [tex]\(\theta\)[/tex] and the product of 74 and the squared sine of [tex]\(\beta\)[/tex]:
[tex]\[
\tan^2(\theta) + 74 \sin^2(\beta)
\][/tex]
5. Set the sum equal to the given value:
According to the equation, these terms together equal 34:
[tex]\[
\tan^2(\theta) + 74 \sin^2(\beta) = 34
\][/tex]
### Interpretation
This equation shows a relationship between the tangent of [tex]\(\theta\)[/tex] and the sine of [tex]\(\beta\)[/tex]. To solve for either [tex]\(\theta\)[/tex] or [tex]\(\beta\)[/tex], further information or additional equations are generally needed. However, without loss of generality, if either angle [tex]\(\theta\)[/tex] or [tex]\(\beta\)[/tex] is specified, you can solve for the other angle based on the context of the problem. The equation provides a constraint that both angles must satisfy.
### Verification
To verify the equation or find specific values of [tex]\(\theta\)[/tex] and [tex]\(\beta\)[/tex] that satisfy it, we can plug in possible angle values until the equation holds true. However, solving an equation in two variables typically requires either additional constraints or context for a specific solution.
Therefore, the given equation:
[tex]\[
\tan^2 \theta + 74 \sin^2 \beta = 34
\][/tex]
is the result of the specified trigonometric relationship between [tex]\(\theta\)[/tex] and [tex]\(\beta\)[/tex]. The equation is fundamental unless further details or constraints are provided to find particular values of [tex]\(\theta\)[/tex] and [tex]\(\beta\)[/tex].
