Question

Given the following right triangle, find cose, sine, tane sececsce, and cote. Do not

approximate: Find exact answers. Show all of your work and explain steps as necessary.​

Answer 1

Explanation:

Use the Pythagorean theorem:

`leg^2+leg^2=hypotenuse^2`

We have

`leg=4,\ hypotenuse=10`

Substitute:

`4^2+leg^2=10^2`

`16+leg^2=100` subtract 16 from both sides

`leg^2=84\to leg=√(84)\\\\leg=√((4)(21))\\\\leg=\sqrt4\cdot√(21)\\\\leg=2√(21)`

`sine=(opposite)/(hypotenuse)\\\\cosine=(adjacent)/(hypotenuse)\\\\tangent=(opposite)/(adjacent)\\\\cotangent=(adjacent)/(opposite)\\\\secant=(hypotenuse)/(adjacent)\\\\cosecant=(hypotenuse)/(opposite)`

Substitute:

`hypotenuse=10,\ opposite=4,\ adjacent=2√(21)`

`\sin\theta=(4)/(10)=(2)/(5)\\\\\cos\theta=(2√(21))/(10)=(√(21))/(5)\\\\\tan\theta=(4)/(2√(21))=(2)/(√(21))\cdot(√(21))/(√(21))=(2√(21))/(21)\\\\\cot\theta=(2√(21))/(4)=(√(21))/(2)\\\\\sec\theta=(10)/(2√(21))=(5)/(√(21))\cdot(√(21))/(√(21))=(5√(21))/(21)\\\\\csc\theta=(10)/(4)=(5)/(2)`