Download 1. You are esitmating

Lesson 5: Solving Right Triangles
Standard: MCC9‐12.G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of
trigonometric ratios for acute angles. MCC9‐12.G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. MCC9‐12.G.SRT.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Essential Question: How do you use trigonometric ratios to solve right triangles?
Warm up: Find the sine, cosine, and the tangent of
1.
and
.
2.
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 25°, then
and ratio of the opposite side
to the hypotenuse is always 0.4226 no matter how big or small the triangle.
To solve a right triangle means to know the lengths of all three sides and the measures of all three angles. In a previous lesson we to
solve right triangles using the special right triangles formed from having either angle measures on 30°-60°-90° or 45°-45°-90°. What
happens when the angles measures are not these special measures? You can solve a right triangle if you know either of the
following: one side length and one acute angle, and two side lengths.

Solving Right Triangles When Given One Side Length and One Acute Angle
1. You are esitmating the height of a Sitka spruce tree in Alaska. You stand 45 feet from the base of the tree. The measure of the
a.
angle a point on the ground to the top of the tree is 59°.
Identify the trigonometric ratio that involves the height h and the known length of 45 feet.
b.
Set up an equation that uses the trigonometric ratio found in a.
c.
Solve the equation from b for the unknown variable.
d.
What is the approximate height of the tree?
2. The escalator at the Wilshire/Vermont Metro Rail Station in Los Angeles rises 76 feet at a 30° angle. To find the distance d a
person travels on the escalator stairs, you can write a trigonometric ratio that involves the hypotenuse and the known leg
length of 76 feet.
3. 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?
Homework:
Solve the following right triangles
2.
1.
3.
4.
Use the given information about
5.
and BC=7
7.
with
to find the unknown side lengths and angle.
6.
and AC=24
and AB=7
Application problems
9. A ramp elevates at 10 degrees to a door that is 3 ft. above
the sidewalk level. What is the length of the ramp to the
nearest tenth of a foot?
8.
, and BC=12
10. A waterslide elevates about 13° from the base to the top of
the waterslide.The slide extends horizontally about 58.2
meters. Estimate the height h of the slide.
Lesson 5: Solving Right Triangles
Standard: MCC9‐12.G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of
trigonometric ratios for acute angles. MCC9‐12.G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. MCC9‐12.G.SRT.8
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.
Essential Question: How do you use trigonometric ratios to solve right triangles?
Warm up: Solve the following right triangle
 Solving Right Triangles When Given Two Side Lengths
In the previous concept we used the trigonometric functions sine, cosine and tangent to find the ratio of particular sides in a right
triangle given an angle. In some cases, you may know the value of a trigonometric ratio and want to know the measure of the
associated angle.
Using the figure at the right, answer the following questions:
1. What do you need to find in order to “solve the right triangle”?
2.
What is the length of AB?
We can use trigonometric ratios to find the
3. Find the sine, cosine, and tangent of
and
?
.
4.
Find the sine, cosine, and tangent of
?
Just like addition and subtraction are inverse operations because one operation undoes the other, we need an inverse operation to
help us solve the equations set up in #3 and #4. For the trigonometric ratios sine, cosine, and tangent the inverses of these
−1
−1
−1
functions, sin , cos , and tan , are used to find the angle measure when the ratio of the side lengths is known.
When we type
into our calculator, the calculator goes to a table and finds the trig ratio associated with 30°, which is 0.5.
When we use an inverse function we tell the calculator to look up the ratio and given us the angle measure. For example,
nd
-1
. On your calculator you would press 2 SIN to get SIN ( and type in 0.5, close the parenthesis and press ENTER.
-1
Your calculator screen should read SIN (0.5) when you press ENTER.
In general, for acute angle A:



if
if
if
5.
What is the measure of
, then
, then
, then
and
?
Solve the following right triangles
1.
3.
5.
2.
4.
6.
7. A ladder leans against a wall and reaches a point 15 ft. up the wall. The base of the ladder is 3.9 ft. from
the wall. To the nearest tent of a foot, what is the length of the ladder? To the nearest degree, what
angle does the ladder make with the wall and the ground?
8.
A shelf extends perpendicularly 24 cm from a wall. You want to place a 28 cm brace under the shelf. To the nearest tenth of a
centimeter, how far below the shelf will the brace be attached to the wall? To the nearest degree, what angle will the brace
make with the shelf and the wall?
9.
A 7.2 meter guy wire goes from the top of a utility pole on the ground that is 3 meters from the base of the pole. To the nearest
tenth of a meter, how tall is the utility pole? To the nearest degree, what angle does the guy wire make with the pole and with
the ground?