Question

Law of sines and law of cosines

Answer 1

4.
`\displaystyle 38 ≈ x`

3.
`\displaystyle 75° ≈ m∠X`

2.
`\displaystyle 52 ≈ x`

1.
`\displaystyle 15° ≈ m∠X`

Explanation:

Before we begin, here are the formulas for both laws:

Solving for Angles

`\displaystyle (x^2 + y^2 - z^2)/(2xy) = cos∠Z \\ (x^2 - y^2 + z^2)/(2xz) = cos∠Y \\ (-x^2 + y^2 + z^2)/(2yz) = cos∠X`

Do not forget to use
`\displaystyle cos^(-1)`in the end or you will throw the result off!

Solving for Edges

`\displaystyle y^2 + x^2 - 2yx\:cos∠Z = z^2 \\ z^2 + x^2 - 2zx\:cos∠Y = y^2\\ z^2 + y^2 - 2zy\:cos∠X = x^2`

Take the square root of the end result or you will throw it off!

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Solving for Angles

`\displaystyle (sin∠Z)/(z) = (sin∠Y)/(y) = (sin∠X)/(x)`

Do not forget to use
`\displaystyle sin^(-1)`in the end or you will throw the result off!

Solving for Edges

`\displaystyle (z)/(sin∠Z) = (y)/(sin∠Y) = (x)/(sin∠X)`

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4.
`\displaystyle (15)/(sin\:22) = (x)/(sin\:110) \hookrightarrow (15sin\:110)/(sin\:22) = x \hookrightarrow 37,627178911... = x \\ \\ 38 ≈ x`

3.
`\displaystyle (-19^2 + 17^2 + 14^2)/(2[17][14]) = cos∠X \hookrightarrow (-361 + 289 + 196)/(476) = cos∠X \hookrightarrow 0,2605042016... = cos∠X; 74,900018217...° = cos^(-1)\:0,2605042016... \\ \\ 75° ≈ m∠X`

2.
`\displaystyle 16^2 + 42^2 - 2[42][16]\:cos\:120 = x^2 \hookrightarrow 256 + 1764 - 1344\:cos\:120 = x^2 \hookrightarrow √(2692) = √(x^2); 2√(673)\:[or\:51,884487084...] = x \\ \\ 52 ≈ x`

You could have also used the Law of Sines sinse the base angles are congruent [30°].

1.
`\displaystyle (sin\:148)/(25) = (sin∠X)/(12) \hookrightarrow (12sin\:148)/(25) = sin∠X \hookrightarrow 0,2543612468... = sin∠X; 14,735739332...° = sin^(-1)\:0,2543612468... \\ \\ 15° ≈ m∠X`

I am joyous to assist you at any time.

Answer 2

Explanation:

`1.\\ \\ (sin148)/(25)=(sinx)/(12)\\ \\ x=sin^(-1)(12sin148)/(12)\\ \\ x=32^o\\ \\ 2.\\ \\ x^2=42^2+16^2-2(42)16cos120\\ \\ x^2=2692\\ \\ x=√(2692)\\ \\ 3.\\ \\ 19^2=17^2+14^2-2(17)14cosx\\ \\ x=cos^(-1)(124)/(476)\\ \\ x=74.9^o\\ \\ 4.\\ \\ (sin22)/(15)=(sin(180-48-22)/(x)\\ \\ x=(15sin110)/(sin22)\\ \\ x\approx 37.63`