Understanding the Trigonometric Expression
The question asks us to find the number of values for the variable $\theta$ that satisfy the given equation:
$$ \sin^2 \theta + \cos^2 \theta - 1 = 0 $$
To solve this, we need to recall a fundamental identity in trigonometry.
Applying the Pythagorean Identity
One of the most basic and important trigonometric identities is the Pythagorean identity, which states:
$$ \sin^2 \theta + \cos^2 \theta = 1 $$
This identity holds true for any real value of $\theta$.
Simplifying the Equation
Now, let's substitute the value of $\sin^2 \theta + \cos^2 \theta$ from the identity into the given equation:
Original Equation: $ \sin^2 \theta + \cos^2 \theta - 1 = 0 $
Substitute using the identity: $ (1) - 1 = 0 $
Simplify the result: $ 0 = 0 $
Interpreting the Result
The equation simplifies to $0 = 0$. This is a true statement, regardless of the specific value chosen for $\theta$. This means the original expression is not an equation with a limited number of solutions, but rather a trigonometric identity that is true for all possible values of $\theta$.
- Since $\theta$ can be any real number (representing angles in degrees or radians), and the identity holds true for every single one of them, there isn't just one, two, or a finite number of solutions.
- The set of all possible values for $\theta$ is infinite.
Conclusion on Theta Values
Because the equation $ \sin^2 \theta + \cos^2 \theta - 1 = 0 $ simplifies to $0 = 0$, which is always true, the equation is satisfied by infinitely many values of $\theta$.
Similar Questions
- If $8 \tan A = 5$, what is the value of $\frac{8\sin A - 7\cos A}{8\sin A + 11\cos A}$?
- If $tan \theta + 3 cot \theta = 2\sqrt{3}$, $0° < \theta < 90°$, then what is the value of $\theta$?
- Find the value of $\frac{\cos 25^\circ - \sin 65^\circ}{\cos 25^\circ + \sin 65^\circ}$
- If $\sin(5x - 40^\circ) = \cos (5y + 40^\circ)$, then the value of $x + y$ is:
Evaluate. $(\frac{\sin 23^\circ \cos 67^\circ+\cos 23^\circ \sin 67^\circ}{\text{cosec}^2 15^\circ- \tan^2 75^\circ})^3$
- The sides of a right-angled triangle, right-angled at B, are 6, 8 and 10 units. C is the vertex opposite to the side with length 8 units. What is the value of $\tan^2 A + \cos^2 C$?
- If $\mu = 60^\circ$, then $\sin\mu + \cos(90^\circ - \mu) = $
- Find the value of (sin $75^\circ$ + sin $15^\circ$).
Evaluate the following.
$$\frac{5\cos^2 120^\circ + 4\sec^2 30^\circ - \tan^2 135^\circ} {\sin^2 30^\circ + \cos^2 30^\circ}$$
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Important Questions from Trigonometry
(secθ + tanθ)/(secθ - tanθ) is equal to:
If tan 45°, cot θ then the value of θ, in radians is
ABC is a triangle If sin (A+B)/2 = √3/2, then the value of sin C/2 is
The angles of elevation of the top of a temple, from the foot and the top of a building 30 m high, are 60° and 30° respectively. Then height of the temple is
what is the principal value of \(\sin^{-1} \left( \sin \dfrac{2 \pi}{3} \right)\) ?
