# 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.

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## 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.

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## 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