Question

find the sine, cosine, and tangent of ∠r. simplify your answers and write them as proper fractions, improper fractions, or whole numbers. sin(r) = cos(r) = tan(r) =

Explanation:

Step1: Identify sides relative to ∠R

In right triangle \( PRQ \) (right - angled at \( P \)), for \( \angle R \):

• Opposite side (\( opp \)) to \( \angle R \) is \( PQ = 70 \)
• Adjacent side (\( adj \)) to \( \angle R \) is \( PR = 24 \)
• Hypotenuse (\( hyp \)) is \( RQ \). First, we need to find the length of the hypotenuse using the Pythagorean theorem \( a^{2}+b^{2}=c^{2} \), where \( a = 24 \), \( b = 70 \), and \( c \) is the hypotenuse \( RQ \).

\( RQ=\sqrt{24^{2}+70^{2}}=\sqrt{576 + 4900}=\sqrt{5476} = 74 \)

Step2: Calculate \( \sin(R) \)

The formula for sine of an angle in a right triangle is \( \sin(\theta)=\frac{\text{opposite}}{\text{hypotenuse}} \)
For \( \angle R \), \( \sin(R)=\frac{PQ}{RQ}=\frac{70}{74}=\frac{35}{37} \)

Step3: Calculate \( \cos(R) \)

The formula for cosine of an angle in a right triangle is \( \cos(\theta)=\frac{\text{adjacent}}{\text{hypotenuse}} \)
For \( \angle R \), \( \cos(R)=\frac{PR}{RQ}=\frac{24}{74}=\frac{12}{37} \)

Step4: Calculate \( \tan(R) \)

The formula for tangent of an angle in a right triangle is \( \tan(\theta)=\frac{\text{opposite}}{\text{adjacent}} \)
For \( \angle R \), \( \tan(R)=\frac{PQ}{PR}=\frac{70}{24}=\frac{35}{12} \)

Answer:

\( \sin(R)=\frac{35}{37} \)
\( \cos(R)=\frac{12}{37} \)
\( \tan(R)=\frac{35}{12} \)