
The number of solutions of the equation in the interval .
A.Two
B.Four
C.Zero
D.None of these
Answer
489k+ views
Hint: Here, we have to find the number of solutions. First, we have to solve the given equation to find the number of solutions. Trigonometric equation is an equation involving one or more trigonometric ratios of unknown angles.
Formula Used:
We will use the trigonometric identities and ;
Complete step-by-step answer:
We will first solve the given equation .
By using the trigonometric identity , we get
Multiplying the terms, we get
Subtracting the like terms, we get
Rewriting the above equation, we get
Above equation is a quadratic equation, so we will factorize the equation to find the value of .
Factorizing by grouping terms, we get
Now factoring out the common term, we get
Using zero product property, we get
Or
By using trigonometric identity , we get
Or
We know that lies between .
Since the values of does not lie between and 1, so, there is no solution.
Therefore, the number of solutions of the equation in the interval is zero.
Hence, the correct option is option C.
Note: We know that trigonometric equations are expressed as ratios of sine, cosine, tangent, cotangent, secant, cosecant angles. All possible values that satisfy the given trigonometric equation are called solutions of the given trigonometric equation. For a complete solution, “all possible values” satisfying the equation must be obtained. When we try to solve a trigonometric equation, we try to find out all sets of values of , which satisfy the given equation.
Formula Used:
We will use the trigonometric identities
Complete step-by-step answer:
We will first solve the given equation
By using the trigonometric identity
Multiplying the terms, we get
Subtracting the like terms, we get
Rewriting the above equation, we get
Above equation is a quadratic equation, so we will factorize the equation to find the value of
Factorizing by grouping terms, we get
Now factoring out the common term, we get
Using zero product property, we get
Or
By using trigonometric identity
Or
We know that
Since the values of
Therefore, the number of solutions of the equation
Hence, the correct option is option C.
Note: We know that trigonometric equations are expressed as ratios of sine, cosine, tangent, cotangent, secant, cosecant angles. All possible values that satisfy the given trigonometric equation are called solutions of the given trigonometric equation. For a complete solution, “all possible values” satisfying the equation must be obtained. When we try to solve a trigonometric equation, we try to find out all sets of values of
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