A given tangent ratio tells you how the two legs of a right triangle compare, but it doesn't give the hypotenuse. If the legs match a familiar right-triangle triple, use it to complete the side ratio without a calculator. For sine, which means opposite over hypotenuse, name the opposite leg from the angle asked for. Keeping the first angle's opposite leg is the tempting slip.
Hints
- Hint 1
A tangent ratio compares the leg across from an angle with the leg next to it. From , which leg is across from the angle, and which leg is next to it?
- Hint 2
A side ratio doesn't fix the actual lengths. A -- right triangle can be scaled up or down, so what is the hypotenuse in the same ratio units as the two legs?
- Hint 3
The opposite leg changes when you move from to . For sine, put the leg opposite over the hypotenuse, not the leg opposite .
Step-by-step
Use the 3-4-5 side ratio
Step 1Read tangent as a leg ratio
No Desmos needed. The -- triple gives the missing side ratio exactly.
Because is the right angle, is the hypotenuse, the side across from it. Tangent means the leg opposite an angle divided by the leg next to it. From , those legs are and :
So describes a ratio, not necessarily the legs' actual lengths.
- Step 2
Complete the side ratio
A -- right triangle has legs and and hypotenuse . Scaling a triangle changes all its side lengths together, so the ratio here is:
- Step 3
Find sine from angle Q
From , is the opposite leg; was opposite . Sine means opposite over hypotenuse, so:
The value of is . Choice B.