Question

Найдите cos(x - п/4) если tan(x)=2 0

Answer 1

Будем считать, что дан tan(x)=2,0.

Находим косинус угла:cos х = 1/√(1 + tg²a) = 1/(√1 + 4) = 1/√5 = √5/5.

Для определения cos(x - п/4) используем формулу вычитания.

cos(x - (π/4)) = cos x*cos((π/4) + sin x*sin((π/4) =

                    = cos x*(√2/2) + sin x*(√2/2) = (√2/5)*(cos x + sin x).

Находим sin x = √(1 + cos²x) = √(1 - (1/5)) = √(4/5) = 2/√5 = 2√5/5.

Тогда  cos(x - (π/4)) = (√2/2)*((√5/5) + (2√5/5) =  (√2/2)*(3√5/5) =

                               = 3 (√10/10).

Answer 2

To find cos(x - π/4) given that tan(x) = 20, we can use the trigonometric identity:

cos(x - π/4) = cos(x)cos(π/4) + sin(x)sin(π/4)

First, let's find the values of cos(x) and sin(x) using the given value of tan(x):

tan(x) = sin(x) / cos(x) = 20

From this, we can deduce that sin(x) = 20 cos(x).

Now, let's substitute the values into the trigonometric identity:

cos(x - π/4) = cos(x)cos(π/4) + sin(x)sin(π/4) = cos(x)(√2/2) + sin(x)(√2/2) = (cos(x) + sin(x))√2/2 = (20 cos(x) + 20 sin(x))√2/2 = (20(cos(x) + sin(x)))√2/2

Since we have the relationship sin(x) = 20 cos(x), we can substitute this in the above expression:

cos(x - π/4) = (20(cos(x) + 20 cos(x)))√2/2 = (20(2 cos(x)))√2/2 = 40 cos(x)√2/2 = 20 cos(x)√2

Since we know that sin^2(x) + cos^2(x) = 1, we can solve for cos(x):

tan^2(x) + 1 = (sin^2(x) / cos^2(x)) + 1 = 1 / cos^2(x) 1 / cos^2(x) = 400 + 1 1 / cos^2(x) = 401 cos^2(x) = 1 / 401 cos(x) = ± √(1 / 401)

However, since we know that tan(x) = 20, which is positive, we can conclude that cos(x) should also be positive. Therefore:

cos(x) = √(1 / 401)

Substituting this back into our previous expression for cos(x - π/4):

cos(x - π/4) = 20 cos(x)√2 = 20 * √(1 / 401) * √2 = 20√(2 / 401)

Therefore, cos(x - π/4) is equal to 20√(2 / 401).