Question

1)cos425°;2)cos2alpha,sin alpha=4/5;3)cos2x×cos x=sin2x×sin x

Answer 1

1)cos425⁰=cos(360⁰+65⁰)=cos(2π+65⁰)=cos65⁰;
2)cos2α=cos²α-sin²α=1-2sin²α=1-2(4/5)²=1-2·16/25=1-32/25=-7/25;
3)cos2x·cosx=sin2x·sinx ⇒cos2x·cosx-sin2x·sinx=0 ⇒
cos(2x+x)=0;⇒cos3x=0 ⇒
3x=π/2+kπ;k∈Z
x=π/6+kπ/3;k∈Z

Answer 2

1) cos(425°):

To find the value of cos(425°), we need to convert the angle to a more familiar range. The cosine function has a periodicity of 360°, meaning that the cosine of an angle is the same as the cosine of that angle plus or minus any multiple of 360°.

So, let's find an equivalent angle within the range of 0° to 360°:

425° = 360° + 65°

Now, we can find the cosine of 65°. In trigonometry, there are often reference angles that we can use to determine the value of trigonometric functions. The reference angle is the acute angle formed between the terminal side of the given angle and the x-axis.

To find the reference angle for 65°, we subtract it from 90°:

90° - 65° = 25°

Now, we have a reference angle of 25°. We know that cosine is positive in the first and fourth quadrants. Since 65° is in the second quadrant, we need to consider the negative value of cosine. So, the cosine of 65° is equal to the negative cosine of the reference angle, which is cos(25°):

cos(65°) = -cos(25°)

To find the value of cos(25°), you can use a scientific calculator or trigonometric tables. The value of cos(25°) is approximately 0.9063.

Therefore, cos(425°) is approximately -0.9063.

2) cos(2α), sin(α) + cos(α) = 4/5:

Let's solve the equation sin(α) + cos(α) = 4/5 for α.

First, square both sides of the equation to eliminate the square root:

(sin(α) + cos(α))^2 = (4/5)^2

Expanding the left side of the equation using the identity (a + b)^2 = a^2 + 2ab + b^2:

sin^2(α) + 2sin(α)cos(α) + cos^2(α) = 16/25

Using the Pythagorean identity sin^2(α) + cos^2(α) = 1:

1 + 2sin(α)cos(α) = 16/25

Rearranging the equation:

2sin(α)cos(α) = 16/25 - 1

2sin(α)cos(α) = 16/25 - 25/25

2sin(α)cos(α) = -9/25

Multiplying both sides by 25 to eliminate the denominator:

50sin(α)cos(α) = -9

Using the identity sin(2α) = 2sin(α)cos(α), we can rewrite the equation:

sin(2α) = -9/50

Therefore, sin(2α) = -9/50.

Now, we need to find cos(2α) using the identity cos^2(2α) + sin^2(2α) = 1:

cos^2(2α) + (-9/50)^2 = 1

cos^2(2α) + 81/2500 = 1

cos^2(2α) = 1 - 81/2500

cos^2(2α) = 2419/2500

Taking the square root of both sides:

cos(2α) = ± sqrt(2419/2500)

Since cos(2α) can be positive or negative, we have two possibilities:

cos(2α) = sqrt(2419/2500) or cos(2α) = -sqrt(2419/2500)

Therefore, cos(2α) = ± sqrt(2419/2500).

3) cos(2x) × cos(x) = sin(2x) × sin(x):

Let's solve the equation cos(2x) × cos(x) = sin(2x) × sin(x) for x.

Using the identity sin(2x) = 2sin(x)cos(x):

cos(2x) × cos(x) = 2sin(x)cos(x) × sin(x)

Dividing both sides by cos(x):

cos(2x) = 2sin(x)sin(x)

cos(2x) = 2sin^2(x)

Using the identity cos(2x) = 1 - 2sin^2(x):

1 - 2sin^2(x) = 2sin^2(x)

Combining like terms:

1 = 4sin^2(x)

Dividing both sides by 4:

sin^2(x) = 1/4

Taking the square root of both sides:

sin(x) = ± sqrt(1/4)

Since sin(x) can be positive or negative, we have two possibilities:

sin(x) = sqrt(1/4) or sin(x) = -sqrt(1/4)

Therefore, sin(x) = ± 1/2.

To find the values of x, we need to consider the unit circle and the quadrants where sin(x) is positive or negative. In the first and second quadrants, sin(x) is positive, so we have:

sin(x) = 1/2

Taking the inverse sine (sin^(-1)) of both sides:

x = sin^(-1)(1/2)

Using a scientific calculator or trigonometric tables, we find that sin^(-1)(1/2) is equal to 30°.

Therefore, one solution is x = 30°.

In the third and fourth quadrants, sin(x) is negative, so we have:

sin(x) = -1/2

Taking the inverse sine (sin^(-1)) of both sides:

x = sin^(-1)(-1/2)

Using a scientific calculator or trigonometric tables, we find that sin^(-1)(-1/2) is equal to -30° or -150°.

Therefore, the other solutions are x = -30° and x = -150°.

To summarize, the solutions to the equation cos(2x) × cos(x) = sin(2x) × sin(x) are x = 30°, x = -30°, and x = -150°.