Answer:
sin (Q) = 15/17
sin (R) = 8/17
cos (Q) = 8/17
cos (R) = 15/17
tan (Q) = 15/8
tan (R) = 8/15
Explanation:
Step 1: Find the length of side QR (i.e., the hypotenuse):
- Because the sine and cosine ratios require us to use the hypotenuse, we first need to find it.
- Since this is a right triangle, we can find the hypotenuse using the Pythagorean Theorem, which is given by:
a^2 + b^2 = c^2, where
- a and b are the triangle's shortest sides called legs,
- and c is the longest side called the hypotenuse.
Thus, we can plug in 16 and 30 for a and b to find x, the hypotenuse (aka the length of side QR):
16^2 + 30^2 = c^2
256 + 900 = c^2
1156 = c^2
√1156 = √1156
34 = c
Thus, the length of side QR (the hypotenuse) is 34 units.
Step 2: Find sin Q and sin R:
sin Q:
The sine ratio is given by sin (θ) = opposite / hypotenuse, where
- θ is the reference angle.
- When angle Q is the reference angle, SR is the opposite side and QR is the hypotenuse.
Thus, sin (Q) = 30/34. This simplifies to sin (Q) = 15/17.
sin R:
When angle R is the reference angle, QS is the opposite side and QR is the hypotenuse.
Thus sin (R) = 16/34. This simplifies to sin(R) = 8/17.
Step 3: Find cos Q and cos R:
The cosine ratio is given by:
cos (θ) = adjacent / hypotenuse, where
- θ is the reference angle.
cos Q:
- When angle Q is the reference angle, QS is the adjacent side and QR is the hypotenuse.
Thus cos (Q) = 16/34. This simplifies to cos (Q) = 8/17.
cos R:
- When angle R is the reference angle, SR is the adjacent side and QR is the hypotenuse.
Thus, cos (R) = 30/34. This simplifies to cos (R) = 15/17.
Step 4: Find tan Q and tan R:
The tangent ratio is given by:
tan (θ) = opposite / adjacent, where
- θ is the reference angle.
tan (Q):
- When angle Q is the reference angle, SR is the opposite side and QS is the adjacent side.
Thus tan (Q) = 30/16. This simplifies to tan (Q) = 15/8.
tan (R):
- When angle R is the reference angle, QS is the opposite side and SR is the adjacent side.
Thus, tan (R) = 16/30. This simplifies to tan (R) = 8/15.