In a right triangle sin (2x°)=cos (3x-10)°. What is the value of x?
To solve this equation, we can use the trigonometric identity: sin(x) = cos(90° - x).
Let's substitute the values we have:
sin(2x°) = cos(3x - 10)°
Using the trigonometric identity, we can rewrite the equation as:
sin(2x°) = sin(90° - (3x - 10)°)
Now, we set the angles equal to each other:
2x° = 90° - (3x - 10)°
Solving for x:
2x° = 90° - 3x + 10°
Combining like terms:
5x° = 100°
Dividing both sides by 5:
x° = 20°
Therefore, the value of x is 20 degrees.
To solve this equation trigonometric equation, we need to simplify it and find the value of x that satisfies the equation.
Let's start by using the trigonometric identity: sin(90° - θ) = cos(θ). This identity will help us simplify the equation.
The given equation is sin(2x°) = cos(3x - 10)°. Applying the identity, we can rewrite the equation as sin(90° - (3x - 10)°) = cos(3x - 10)°.
Now, we can simplify the equation further.
sin(90° - (3x - 10)°) = cos(3x - 10)°
Using the property of subtraction, we have:
sin(90° - 3x + 10°) = cos(3x - 10)°
Simplifying the inside brackets:
sin(100° - 3x) = cos(3x - 10)°
Now, using the trigonometric identity sin(θ) = cos(90° - θ), we can rewrite the equation as:
cos(90° - (100° - 3x)) = cos(3x - 10)°
cos(-10° + 3x) = cos(3x - 10)°
Since cos(θ) = cos(θ), we can equate the angles:
-10° + 3x = 3x - 10
-10° + 10 = 3x - 3x
0 = 0
The equation simplifies to 0 = 0, which is true for all values of x.
Therefore, the original equation sin(2x°) = cos(3x - 10)° has infinite solutions for x. Any value of x will satisfy the equation.
by the complimentary property:
cos (3x-10)°
= sin(90 - (3x-10) )
= sin (100 - 3x)
so sin(2x) = sin(100-3x)
2x = 100-3x
5x = 100
x = 20
check:
sin (2x) = sin 40°
cos(3x-10) = cos 50°
indeed sin 40 = cos 50