Precalculus : Trigonometric Applications

Study concepts, example questions & explanations for Precalculus

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Example Questions

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Example Question #1 : Trigonometric Applications

In a right triangle, if the hypotenuse is  and a leg is , what is the area of the triangle? 

Possible Answers:

Correct answer:

Explanation:

Use the Pythagorean Theorem to find the other leg.

The length of the given leg is 3, and the unknown leg is .

Use the area of a triangle formula and solve.

Example Question #1 : Solving Right Triangles

An isosceles right triangle has a hypotenuse of 1.  What is the area of this triangle?

Possible Answers:

Correct answer:

Explanation:

Write the formula for the Pythagorean theorem.

In an isosceles right triangle, both legs of the right triangle are equal.  

Substitute the either variable and the known hypotheuse and determine the side length.

This length represents both the base and the height of the triangle. Write the area of a triangle and substitute to solve for the area.

 

Example Question #1 : Trigonometric Applications

Solve the right triangle.

Pcq1

Solve the right triangle given that a=5, b=12, and A=22.620°

Possible Answers:

B=67.380°

C=90°

c=17

B=67.380°

C=90°

c=13

B=90°

C=67.380°

c=13

None of these answers are correct.

B=90°

C=67.380°

c=169

Correct answer:

B=67.380°

C=90°

c=13

Explanation:

Pcq1

C is given as 90°.

A is given as 22.620°

a is given as 5

b is given as 12

 

Therefore...

 

All angles of a triangle add up to equal 180°.

Example Question #1 : Trigonometric Applications

Solve the right triangle.

Pcq1

C=90°

B=45°

a=5

c=

Possible Answers:

A=45°

b=

A=135°

b=5

A=135°

b=2.07

None of these answers are correct.

A=45°

b=5

Correct answer:

A=45°

b=5

Explanation:

Pcq1

Given that:

C=90°

B=45°

a=5

c=

 

Therefore...

 

All angles of a triangle add up to 180°.

Example Question #2 : Trigonometric Applications

A right triangle has a base of 10 and a hypotenuse of 20.  What is the length of the other leg?

Possible Answers:

Correct answer:

Explanation:

Write the Pythagorean Theorem.

Substitute the values of the leg and hypotenuse.  The hypotenuse is the longest side of the right triangle.  Solve for the unknown variable.

Example Question #1 : Solve A Right Triangle

In the right triangle ABC, side AB is  cm long, side AC is  cm long, and side BC is the hypotenuse. How long is side BC?

Possible Answers:

 cm

 cm

 cm

 cm

Correct answer:

 cm

Explanation:

Given that ABC is a right triangle, the length of hypotenuse BC is the root of the sum of the squares of the two other sides (in other words, . Since AB is  cm long and AC is  cm long, we get that , and so .

Example Question #1 : Trigonometric Applications

Isoscelestriangle_800

Given , and the lower angles of the isosceles triangle are , what is the length of ? Round to the nearest tenth.

Possible Answers:

Correct answer:

Explanation:

Since the angle of the isosceles is , the larger angle of the right triangle formed by  is also .

Using , we can find 

.

Then solve for 

.

Simplify: .

Lastly, round and add appropriate units: .

Example Question #1 : Trigonometric Applications

In isosceles triangle , .  If side , what is the approximate length of the two legs  and ?

Possible Answers:

Correct answer:

Explanation:

Isos

In the diagram, AB is cut in half by the altitude. 

From here it easy to use right triangle trigonometry to solve for AC.

Example Question #1 : Solving Right Triangles

Find the area of the following Isosceles triangle (units are in cm):

Varsity log graph

Possible Answers:

Correct answer:

Explanation:

The formula for the area of a triangle is:

 

We already know what the base is and we can find the height by dividing the isosceles triangle into 2 right triangles: 

 

Varsity log graph

From there, we can use the Pathegorean Theorem to calculate height:

To find the area, now we just plug these values into the formula:

Example Question #1 : Trigonometric Applications

Find the area of the given isosceles triangle and round all values to the nearest tenth:

Varsity log graph

Possible Answers:

Correct answer:

Explanation:

The first step to solve for area is to divide the isosceles into two right triangles:

Varsity log graph

From there, we can determine the height and base needed for our area equation  

From there, height can be easily determined using the Pathegorean Theorem:

Now both values can be plugged into the Area formula:

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