Question

I need a step by step explanation please Thank you so much

Answer 1

a. To find where tan 0 = tan 265° and 0 is not equal to 265°, we can use the following formula:

tan(theta) = sin(theta) / cos(theta)

So we have:

tan(0) = tan(265°)

sin(0) / cos(0) = sin(265°) / cos(265°)

Since 0 is not equal to 265°, we know that cos(0) is not equal to cos(265°). Therefore, we can simplify the equation as follows:

sin(0) * cos(265°) = sin(265°) * cos(0)

Using the identity sin(a - b) = sin(a)cos(b) - cos(a)sin(b), we can rewrite this equation as:

sin(0 - 265°) = sin(-265°) = -sin(265°)

Since sin(-x) = -sin(x), we have:

sin(0 + 265°) = sin(265°)

Using the identity sin(a + b) = sin(a)cos(b) + cos(a)sin(b), we can rewrite this equation as:

sin(0)cos(265°) + cos(0)sin(265°) = sin(265°)

Since tan(theta) = sin(theta)/cos(theta), we can divide both sides of the equation by cos(0):

tan(0) + tan(265°) = 1

tan(0) + (-1.1918...) = 1

tan(0) ≈ **2.1918...**

Therefore, the solution is **tan 0 ≈ 2.1918...**.

b. If sin 0 = 2/3 and cos 0 > 0, then we can use the following formula to find cotangent:

cot(theta) = cos(theta)/sin(theta)

We are given that sin 0 = 2/3 and cos 0 > 0, so we know that:

cos^2(theta) + sin^2(theta) = 1

cos^2(theta) + (2/3)^2 = 1

cos^2(theta) = 1 - (2/3)^2

cos^2(theta) = 5/9

Since cos 0 > 0, we know that cos theta is positive. Therefore:

cos(theta) = sqrt(5/9)

= (sqrt(5))/3

Now we can use the formula for cotangent:

cot(0) = cos(0)/sin(0)

= [(sqrt(5))/3] / (2/3)

= sqrt(5)/2

Therefore, the solution is **cot 0 = sqrt(5)/2**.

c. If 5/2 cos 0 +4 =2, we can solve for cos 0 as follows:

5/2 cos 0 +4 =2

5/2 cos 0 = -2

cos 0 = -4/5

Now we can use the inverse cosine function to find the angle:

cos^-1(-4/5)

≈ **131.8°**

Therefore, the solution is **0 ≈ 131.8°**.

I hope this helps! Let me know if you have any other questions.

Answer 2

Answers:

• (a) 85
• (b)
  `\boldsymbol{(√(5))/(2)}`
• (c) Approximately 143.1301 and 216.8699

======================================================

Work shown for part (a)

tan(x) = tan(x-180)

tan(265) = tan(265-180)

tan(265) = tan(85)

-------------------------

Work shown for part (b)

sine = opposite/hypotenuse = 2/3

opposite = 2 and hypotenuse = 3

Use a = 2 and c = 3 to determine b in the pythagorean theorem.

`a^2+b^2 = c^2\\\\2^2+b^2 = 3^2\\\\4+b^2 = 9\\\\b^2 = 9-4\\\\b^2 = 5\\\\b = √(5)\\\\`

adjacent =
`√(5)` and opposite = 2

`\cot(\theta) = \frac{\text{adjacent}}{\text{opposite}}\\\\\cot(\theta) = (√(5))/(2)\\\\`

-------------------------

Work shown for part (c)

`(5)/(2)\cos(\theta)+4 = 2\\\\(5)/(2)\cos(\theta) = 2-4\\\\(5)/(2)\cos(\theta) = -2\\\\\cos(\theta) = -2*((2)/(5))\\\\\cos(\theta) = -(4)/(5)\\\\`

`\theta = \pm\arccos\left(-(4)/(5)\right)+360n \ \ \text{ .... where n is an integer} \\\\\theta = \pm143.1301+360n\\\\\theta = 143.1301+360n \ \text{ or } \ \theta = -143.1301+360n\\\\`

Here's a table of values for selected inputs of n

`\begin{array}c \cline{1-3}n & 143.1301+360n & -143.1301+360n\\\cline{1-3}-1 & -216.8699 & -503.1301\\\cline{1-3}0 & 143.1301 & -143.1301\\\cline{1-3}1 & 503.1301 & 216.8699\\\cline{1-3}2 & 863.1301 & 576.8699\\\cline{1-3}\end{array}`

The results 143.1301 and 216.8699 are in the interval
`0^(\circ) < \theta < 360^(\circ)`, which makes them the two approximate solutions.

You can use graphing software such as GeoGebra or Desmos to confirm the answers.