Question

Help cos(pi/2-t)-sin(pi+t)=√2

Answer 1

$cos(\frac{\pi }{2}-t)-sin(\pi +t)=\sqrt2\\\\sint-(-sint)=\sqrt2\\\\sint+sint=\sqrt2\\\\2sint=\sqrt2\\\\sint=\frac{\sqrt2}{2}\\\\t=(-1)^{n}\cdot \frac{\pi }{4}+\pi n\; ,\; n\in Z$

Answer 2

Решение прикреплено. Удачи

Answer 3

To solve the trigonometric equation cos(pi/2 - t) - sin(pi + t) = √2, we'll first use some trigonometric identities to simplify the left-hand side of the equation. Then, we'll solve for the value of t.

Step 1: Apply trigonometric identities Recall the following trigonometric identities:

1. cos(a - b) = cos(a)cos(b) + sin(a)sin(b)
2. sin(a + b) = sin(a)cos(b) + cos(a)sin(b)

Using these identities, we can rewrite the left-hand side of the equation as follows: cos(pi/2 - t) - sin(pi + t) = cos(pi/2)cos(t) + sin(pi/2)sin(t) - sin(pi)cos(t) - cos(pi)sin(t)

Since cos(pi/2) = 0 and sin(pi/2) = 1, and cos(pi) = -1 and sin(pi) = 0, the equation becomes: 0cos(t) + 1sin(t) - (-1)cos(t) - 0sin(t) = 2sin(t) + cos(t)

Step 2: Now the equation is simplified to: 2sin(t) + cos(t) = √2

Step 3: Rewrite √2 as (√2/2)^2 Now, rewrite √2 as (√2/2)^2, since sin(45 degrees) = cos(45 degrees) = √2/2.

2sin(t) + cos(t) = (√2/2)^2

Step 4: Combine like terms 2sin(t) + cos(t) = 2/2

Step 5: Simplify further 2sin(t) + cos(t) = 1

Step 6: Solve for t To solve for t, we need to find the values of t that satisfy the equation 2sin(t) + cos(t) = 1. There might be multiple solutions for t.

One way to solve this is to notice that 2sin(t) + cos(t) is similar to the cosine addition formula: cos(a + b) = cos(a)cos(b) - sin(a)sin(b).

So, we can rewrite the equation as follows: cos(t + π/4) = 1

Now, for the cosine function to be equal to 1, its angle should be 0 degrees or any multiple of 2π. So, we have: t + π/4 = 2πn, where n is an integer.

Now, solve for t: t = 2πn - π/4

This will give you all possible solutions for t that satisfy the original equation. Keep in mind that there are infinitely many solutions because n can take any integer value.