Trigonometry

To convert degrees to radians, you can use the following formula:

Radians = Degrees × (π / 180)

So, for 40 degrees:

Radians = 40 × (π / 180)

Radians = 2π / 9

Radians ≈ 0.698

Therefore, 40 degrees is approximately 0.698 radians.

For more information, you can use online converters. Here's a link to a Google Search

To determine the quadrant in which an angle of -1560 degrees lies, we first need to find its coterminal angle within the range of 0 to 360 degrees.

We can do this by adding multiples of 360 to -1560 until we get an angle in the desired range:

-1560 + 360 = -1200

-1200 + 360 = -840

-840 + 360 = -480

-480 + 360 = -120

-120 + 360 = 240

So, -1560 degrees is coterminal with 240 degrees.

Now we determine the quadrant:

• Quadrant I: 0° to 90°
• Quadrant II: 90° to 180°
• Quadrant III: 180° to 270°
• Quadrant IV: 270° to 360°

Since 240° is between 180° and 270°, the angle lies in Quadrant III.

Here's how to solve this problem:

1. Solve cos(θ) = sin(θ):

   • This equation is true when θ = 45° + 180° * n, where n is any integer. This is because sine and cosine are equal at 45 degrees, and then repeat this relationship every 180 degrees due to their periodicity and phase difference.

2. Incorporate the Given Information

   • We are given that θ is 83 times greater than 90 degrees, meaning θ > 83 * 90°. θ > 7470°.

3. Find the Appropriate Angle:

   • We need to find an integer n such that θ = 45° + 180° * n and θ > 7470°.

   • Let's solve for n : 45 + 180*n* > 7470

   • 180*n* > 7425

   • n > 41.25

   • The smallest integer n that satisfies this condition is n = 42.

   • Therefore, θ = 45° + 180° * 42 = 45° + 7560° = 7605°.

4. Calculate 60 * θ

   • 60 * θ = 60 * 7605° = 456300°

To find the general solution of the equation cot(x) + tan(x) = 2 csc(x), we can rewrite the equation in terms of sine and cosine.

Recall that cot(x) = cos(x)/sin(x), tan(x) = sin(x)/cos(x), and csc(x) = 1/sin(x). Substituting these into the equation, we get:

cos(x)/sin(x) + sin(x)/cos(x) = 2/sin(x)

Multiplying both sides by sin(x)cos(x) to clear the fractions, we obtain:

cos²(x) + sin²(x) = 2cos(x)

Since cos²(x) + sin²(x) = 1, the equation simplifies to:

1 = 2cos(x)

cos(x) = 1/2

The general solution for cos(x) = 1/2 is given by:

x = 2πn ± π/3, where n is an integer.

We need to check for extraneous solutions. The original equation involves cot(x), tan(x), and csc(x), so we must ensure that sin(x) ≠ 0 and cos(x) ≠ 0.

sin(x) ≠ 0 when x ≠ nπ, where n is an integer.

cos(x) ≠ 0 when x ≠ (2n+1)π/2, where n is an integer.

The solutions x = 2πn ± π/3 do not violate these conditions. Therefore, the general solution is:

x = 2πn + π/3 or x = 2πn - π/3, where n is an integer.