Question

Find sin(2x), cos(2x) and tan (2x) from the given information. cot(x)=2/3, x in quadrant I

Answer 1

To find sin(2x), cos(2x), and tan(2x) from the given information, use the double angle formulas. Substitute the values of cot(x) = 2/3 in the formulas and simplify to find sin(2x) = 6√10/13, cos(2x) = 1/13, and tan(2x) = 6√10.

Step-by-step explanation:

To find the values of sin(2x), cos(2x), and tan(2x) from the given information, we need to use the double angle formulas for trigonometric functions.

Starting with sin(2x), we can use the identity sin(2x) = 2sin(x)cos(x). Since we know that cot(x) = 2/3 and x is in quadrant I, we can substitute the values and find that sin(x) = 3/√13 and cos(x) = √10/√13. Plugging these values into the double angle formula, we get sin(2x) = 2(3/√13)(√10/√13) = 6√10/13.

Similarly, we can find cos(2x) using the identity cos(2x) = cos²(x) - sin²(x). Plugging in the values, we get cos(2x) = (√10/√13)² - (3/√13)² = 10/13 - 9/13 = 1/13.

Finally, to find tan(2x), we can use the identity tan(2x) = sin(2x)/cos(2x). Plugging in the values, we get tan(2x) = (6√10/13)/(1/13) = 6√10.

Answer 2

1.) 6√13/13

2.) 4√13/13

3.) 3

Step-by-step explanation:

given that cot(x) = 2/3

But cot(x) = 1/tan(x)

Substitutes 1/tan(x) for cot(x)

1/tan(x) = 2/3

Reciprocate both sides

Tan(x) = 3/2 = opposite/adjacent

Use pythagorean theorem to find the hypothenus.

Hypothenus = sqrt ( 3^2 + 2^2 )

Hypothenus = sqrt ( 9 + 4 )

Hypothenus = sqrt (13)

Hypothenus = √13

1.) Sin(x) = opposite/hypothenus = 3/√13

Rationalise

3/√13 × √13/√13

3√13/13

Sin(2x) = 2 × 3√13/13

Sin(2x) = 6√13/13

2.) Cos( x ) = adjacent/hypothenus

Cos (x) = 2/√13

Rationalise

2/√13 × √13 /√13

2√13/13

Cos(2x) = 2 × 2√13/13

Cos(2x) = 4√13/13

3.) Tan (x) = 3/2

tan (2x) = 2 × 3/2

Tan(2x) = 3