Question

10-11 класс. Тригонометрия.Срочно нужно.Решить уравнение:а) 3sin(П+x)+2cos(П/2-x)=-1/2;Вычислить:б) sin(-9П)+2cos 25П/3 - ctg 11П/4в) sin2x/1+cosx + cosxг) (sint/1+cost + sint/1-cost) *sintд) (sin10+sin80)*(cos80-cos10)/sin70 (всё что в скобках разделить на sin70, по общей чертой)Буду премного благодарна за решение. Решение подробное должно быть.

Answer 1

1)3sin(П+x)=-3sinx

2cos(П/2-x)=2sinx

-3sinx+2sinx=-1/2

sinx=1/2 дальше сама..

2)sin(-p)=0  2p убираешь,т.к целый оборот 

2cos 25П/3=2cos (6 1/3 p)=2cosП/3=1

ctg 11П/4=ctg(2 3p/4)=ctg(3p/4)=ctg(180-45)=-ctg45=-1

ответ 1-(-1)=0

всё....надоела тригонометрия.....в помощь таблица приведения...может будем сами делать домашку то?

Answer 2

Solving the Equation

To solve the equation 3sin(π+x) + 2cos(π/2-x) = -1/2, we will use trigonometric identities and algebraic manipulations.

Let's start by simplifying the equation step by step:

1. Use the identity sin(π + x) = -sin(x): - The equation becomes -3sin(x) + 2cos(π/2 - x) = -1/2.

2. Use the identity cos(π/2 - x) = sin(x): - The equation becomes -3sin(x) + 2sin(x) = -1/2.

3. Combine like terms: - The equation simplifies to -sin(x) = -1/2.

4. Multiply both sides by -1 to isolate sin(x): - The equation becomes sin(x) = 1/2.

5. Take the inverse sine (arcsin) of both sides to find the value of x: - The equation becomes x = arcsin(1/2).

Now, let's calculate the value of x using a calculator or table of trigonometric values.

The value of arcsin(1/2) is π/6 or approximately 0.5236 radians.

Therefore, the solution to the equation is x = π/6 or approximately 0.5236 radians.

Calculating Trigonometric Expressions

Now, let's calculate the given trigonometric expressions:

a) sin(-9π) + 2cos(25π/3) - ctg(11π/4): - To calculate this expression, we need to find the values of sin(-9π), cos(25π/3), and ctg(11π/4). - Using the periodicity of trigonometric functions, we can simplify the expression as follows: - sin(-9π) = sin(-π) = -sin(π) = 0. - cos(25π/3) = cos(8π + π/3) = cos(π/3) = 1/2. - ctg(11π/4) = 1/tan(11π/4) = 1/tan(π/4) = 1/1 = 1. - Therefore, the expression simplifies to 0 + 2(1/2) - 1 = 1.

b) sin(2x)/(1 + cos(x)) + cos(x): - To calculate this expression, we need the value of x. - From the previous solution, we found that x = π/6. - Substituting this value into the expression, we get: - sin(2(π/6))/(1 + cos(π/6)) + cos(π/6) - Simplifying further: - sin(π/3)/(1 + cos(π/6)) + cos(π/6) - Using the values of sin(π/3) and cos(π/6) from the trigonometric table or calculator, we get: - √3/2 / (1 + √3/2) + √3/2 - Rationalizing the denominator: - (√3/2) / (2/2 + √3/2) + √3/2 - Simplifying further: - (√3/2) / (2 + √3) + √3/2 - Multiplying the numerator and denominator by the conjugate of the denominator: - (√3/2) * (2 - √3) / (2 + √3) * (2 - √3) + √3/2 - Simplifying the denominator: - (√3/2) * (2 - √3) / (4 - 3) + √3/2 - Simplifying further: - (√3/2) * (2 - √3) + √3/2 - Distributing: - (√3/2) * 2 - (√3/2) * √3 + √3/2 - Simplifying: - √3 - (√3/2) + √3/2 - Canceling out like terms: - √3

c) (sin(t)/(1 + cos(t))) + (sin(t)/(1 - cos(t))) * sin(t): - To calculate this expression, we need the value of t. - However, the value of t is not provided in the question. - Please provide the value of t so that we can calculate the expression accurately.

d) (sin(10) + sin(80)) * (cos(80) - cos(10)) / sin(70): - To calculate this expression, we can use the values of sin(10), sin(80), cos(80), cos(10), and sin(70) from the trigonometric table or calculator. - Substituting these values into the expression, we get: - (sin(10) + sin(80)) * (cos(80) - cos(10)) / sin(70) - Simplifying further: - (0.1736 + 0.9848) * (-0.1736 - 0.9848) / 0.9397 - Calculating the numerator and denominator separately: - 1.1584 * (-1.1584) / 0.9397 - Simplifying: - -1.3410

Please provide the value of t for part c) so that we can calculate the expression accurately.