Question

Use the figure below to find the six corresponding trigonometric values. You can express each value as a fraction or as a decimal rounded to the nearest hundredth.right triangle, leg of length 10 opposite angle A, other leg has length of 8, hypotenuse is not known.1. What is sin(A)?2. What is cos(A)?3. What is tan(A)?4. What is csc(A)?5. What is sec(A)?6. What is cot(A)?

Answer 1

Simplifying we get:Now, AWe are given a right triangle and angle "A". We are given the opposite and adjacent sides of the triangle with respect to angle "A". To determine the six trigonometric functions we will first use the Pythagorean theorem to determine the value of the hypotenuse:

`h^2=a^2+b^2`

Where:

`\begin{gathered} h=\text{ hypotenuse} \\ a,b=\text{ sides} \end{gathered}`

Now, we substitute the values:

`h^2=10^2+8^2`

Solving the squares:

`h^2=100+64`

Adding the values:

`h^2=164`

Now, we take the square root to both sides:

`h=√(164)`

We can factor the radical as follows:

Now, we determine the sine of "A". To do that we will use the following definition of sine:

`h=√(41*4)`

Now, we distribute the radical:

`h=√(4)√(41)`

Solving the square root:

`h=2√(41)`

Now, we determine the value of the sine of "A" using the following definition:

`sin\left(A\right)=(opposite)/(hypotenuse)`

Now, we substitute the values:

`sin\left(A\right)=(10)/(2√(41))`

Now, we simplify:

`sin\left(A\right)=(5)/(√(41))`

Now, we determine the cosine of "A". We use the following definition:

`cos\left(A\right)=(adjacent)/(hypotenuse)`

Substituting the values we get:

`cos\left(A\right)=(8)/(2√(41))`

Simplifying we get:

`cos\left(A\right)=(4)/(√(41))`

Now, we determine the value of the tangent of "A" using the following definition:

`tan\left(A\right)=(opposite)/(adjacent)`

Substituting the values we get:

`tan\left(A\right)=(10)/(8)`

Simplifying we get:

`tan\lparen A)=(5)/(4)`

Now, we determine the cosecant using the following definition:

`csc\lparen A)=(1)/(sin\left(A\right))`

This means that the value of the cosecant is the invert of the sine:

`csc\left(A\right)=(√(41))/(5)`

To determine the value of the secant we use the following definition:

`sec\left(A\right)=(1)/(cos\left(A\right))`

Therefore, we invert the value of the cosine:

`sec\left(A\right)=(√(41))/(4)`

the value of the cotangent is determined using the following definition:

`cot\left(A\right)=(1)/(tan\left(A\right))`

This means that we need to invert the value of the tangent, we get:

`cot\left(A\right)=(4)/(5)`

And thus we have determined the trigonometric functions.