An isoceles right triangle has leg lengths of 4 units each. What is the sine of each of the triangle's acute angles?
Guidance
The trigonometric ratios sine, cosine and tangent refer to the known ratios between particular sides in a right triangle based on an acute angle measure.
In this right triangle, side
If we consider the angle
If we consider the angle
Now we can define the trigonometry ratios as follows:
A shorthand way to remember these ratios is to take the letters in red above and write the phrase:
Now we can find the trigonometric ratios for each of the acute angles in the triangle above.
It is important to understand that given a particular (acute) angle measure in a right triangle, these ratios are constant no matter how big or small the triangle. For example; if the measure of the angle is
Example A
Find the trig ratios for the acute angles
Solution:
From angle
From angle
Do you notice any patterns or similarities between the trigonometric ratios? The opposite and adjacent sides are switched and the hypotenuse is the same. Notice how this switch affects the ratios:
Example B
Use trigonometric ratios to find the
Solution:
First identify or label the sides with respect to the given acute angle. So,
To find
NOTE: make sure that your calculator is in DEGREE mode. To check, press the MODE button and verify that DEGREE is highlighted (as opposed to RADIAN). If it is not, use the arrow buttons to go to DEGREE and press ENTER. The default mode is radian, so if your calculator is reset or the memory is cleared it will go back to radian mode until you change it.
To find
Alternatively, we could find
The downside of this method is that if we miscalculated our
Example C
Given
Solution: Visual learners may find it particularly useful to make a sketch of this triangle and label it with the given information:
To find
Concept Problem Revisit
If you draw the triangle described in this problem, you will see that the sine
Let's use the Pythagorean Theorem.
Therefore, the sine of both of the acute angles is
Guided Practice
1. Use trig ratios to find
2. Given
3. The base of a playground slide is 6 ft from the base of the platform and the slide makes a
Answers
1. For
For
2. For
For
3.
Explore More
Use you calculator to find the following trigonometric ratios. Give answers to four decimal places.

sin35∘ 
tan72∘ 
cos48∘ 
tan45∘ 
sin30∘ 
cos88∘  Write the three trigonometric ratios of each of the acute angles in the triangle below.
Use trigonometric ratios to find the unknown side lengths in the triangles below. Round your answers to the nearest hundredth.
For problems 1113 use the given information about

a=12 andm∠A=43∘ 
m∠C=75∘ andb=24 
c=7 andm∠A=65∘  A ramp needs to have an angle of elevation no greater than 10 degrees. If the door is 3 ft above the sidewalk level, what is the minimum possible ramp length to the nearest tenth of a foot?

A ship,
Sea Dancer
, is 10 km due East of a lighthouse. A second ship,
Nelly
, is due north of the lighthouse. A spotter on the
Sea Dancer
measures the angle between the
Nelly
and the lighthouse to be
38∘ . How far apart are the two ships to the nearest tenth of a kilometer?