Trigonometry with Non-Right Triangles - Trigonometry - High School Geometry Unlocked (2016)

High School Geometry Unlocked (2016)

Chapter 4. Trigonometry

Lesson 4.4. Trigonometry with Non-Right Triangles

Sometimes, we know more than we think we do…

What is the area of the triangle above?

In order to find the area, you’ll need to find the lengths of the height and base. First, draw a line representing the height. Label the unknown sides as a, b, and c. The height must always be perpendicular to the base, so you know it forms a right angle. Therefore, the smaller triangle with 45° and 90° must be a 45°-45°-90° triangle.

You can use the given side length of 4 to find more information. 4 is the hypotenuse of our 45°-45°-90° triangle. Therefore, the two legs of this smaller triangle are each equal to 4.

To prove this, recall the “special” right triangle 45°-45°-90° has side lengths x, x, and x.

Or, you can use the Pythagorean theorem: a2 + a2 = (4)2. (The 45°-45°-90° triangle is isosceles).

Or, you can use sin(45°) = .

That’s a lot of options!

height = 4

Now that you know the height of this triangle, you can also find the other unknown side lengths.

If the height is 4, then b is 4, and c must be 8.

To prove this, use the relationships of the “special” right triangle 30°-60°-90°.

Or, use sin, cos, or tan functions to solve. (For example, use tan(60°) = and cos(60°) = ).

base = 4 + 4

 ≈ 10.93

Therefore, the area of the triangle is determined as follows:

A = bh

 ≈ × 4 × (10.93)

 ≈ 2 × (10.93)

 ≈ 21.86

The Law of Sines

Because of the ability to make triangles within triangles, yet another identity can be derived.

The Law of Sines

in which side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C.

The way to use the Law of Sines is to create a proportion. Each angle corresponds with its opposite side.

Let’s take another look at the 105-30-45 triangle.

Using the Law of Sines, we have the following proportion:

Use your calculator to confirm that the proportion is true:

≈ 0.0884

≈ 0.0884

≈ 0.0884

We can use the Law of Sines to solve for unknown side lengths or angles.

What is the value of x?

To solve, use the Law of Sines. Note that in most cases, a proportion can be two fractions instead of three. You don’t always need all three “sides” to the Law of Sines equation.

The Law of Cosines

The Law of Cosines is based on three sides and one angle. It allows you to solve if one of those facts is unknown.

The Law of Cosines

c2 = a2 + b2 − 2ab (cos(C))

in which side a is opposite angle A, side b is opposite angle B, and side c is opposite angle C.

Here is how you may see the Law of Sines and the Law of Cosines on the ACT.

Triangle JKL is shown in the figure below. The measure of ∠K is 50°, JK = 9 cm, and KL = 6 cm. Which of the following is the length, in centimeters, of LJ?

Note: For a triangle with sides of length a, b, and c opposite angles ∠A,B, and ∠C, respectively, the Law of Sines states , and the Law of Cosines states c2 = a2 + b2 − 2ab (cos(C)).

A.9sin 50°

B.6sin 50°

C.

D.

E.

See it in action with a 30°-60°-90° triangle.

The most important
thing to remember
when setting up the
Law of Cosines, is
that the angle C is
opposite side c.

There are a few different ways we can set up the Law of Cosines with this figure.

There are no unknowns in this example, so just simplify and use your calculator to see that the equations are true. We’ll do the first one step-by-step:

That’s about right! If you rounded the irrational numbers, you can expect the answer to be inexact.

What is the value of x?

Solve this using the Law of Cosines. Remember that the angle C is opposite side c in the equation.

The first decision you have to make is which one (or both) of these laws is useful to you. You’re trying to solve for a side where you have the opposite angle, but you don’t have angles to match up with either of the other two sides you know. The sides are not equivalent, so you can’t assume that the angles are equivalent. Consequently, you may not have enough information to use the Law of Sines. The Law of Cosines, on the other hand, would let you solve for the missing side, c, knowing only the other sides and the opposite angle. Line up each piece of the formula to find that LJ2 = 92 + 62 − 2(9)(6)cos 50°. Before you start calculating this value, glance at your answer choices—they aren’t asking you to solve completely, just to match up the filled-in formula. Take the square root of both sides to find LJ = , or (E). The correct answer is (E).

ANSWERS TO CHAPTER 4 EXERCISES

Example 1 Answers

Example 2 Answers

Example 3 Answers

Example 4 Answers

x53.13°

To solve for x°, you can enter sin–1, cos–1, or tan–1.

y36.87°

To solve for y°, you can enter sin–1, cos–1, or tan–1.

DRILL

CHAPTER 4 PRACTICE QUESTIONS

Directions: Complete the following problems as specified by each question. For extra practice after answering each question, try using an alternative method to solve the problem or check your work.

Click here to download a PDF of Chapter 4 Practice Questions.

1.In triangle XYZ, . Determine the values of the 6 trigonometric functions sin, cos, tan, csc, sec, and cot.

2.A window washer is cleaning the windows of a building. The ladder is fully extended to 25 feet and is leaning against the building. The base of the ladder is 7 feet away from the wall of the building.

a.What is the vertical height, on the building, that the ladder reaches?

b.What angle is created with the ground?

3.Given csc θ = , determine the values of the other five trigonometric functions. Rationalize the denominators, if needed.

4.Given the triangle with the labeled side lengths, show that the following are true:

A.sin θ × sec θ = tan θ

b.sin2θ + cos2θ = 1

5.Prove sin (90 − θ) = cos θ.

6.Prove tan (90 − θ) = .

7.For triangle ABC, AB = 5, BC = 7, and m∠A = 43°. Find the exact value of ∠C; then approximate the value to 2 decimal places.

8.Triangle ABC has side lengths of AB = 19, BC = 8, and AC = 14. Use the lengths to find the three angles of the triangle.

SOLUTIONS TO CHAPTER 4 PRACTICE QUESTIONS

1.

The question provides the ratio for two sides of the right triangle, so use the Pythagorean theorem to determine the third side:

a2 + b2 = c2

Pythagorean theorem

(5)2 + b2 = (13)2

Substitute a = 5 and c = 13.

25 + b2 = 169

Simplify.

  b2 = 144

Subtract 25 from both sides.

  b = 12

Take the square root of both sides.

Note: Recall common Pythagorean triples to save time in calculations!

In terms of θ, the opposite side is 12, the adjacent side is 5, and the hypotenuse is 13. Determine the values of the 6 trigonometric functions:

2.It can be helpful to construct a figure that depicts the scenario taking place:

a.24 feet

To determine the height, use the Pythagorean theorem:

a2 + b2 = c2

Pythagorean theorem

(7)2 + b2 = (25)2

Substitution using a = 7 and c = 25.

49 + b2 = 625

Simplify.

  b2 = 576

Subtract 49 from both sides.

  b = 24

Take the square root of both sides.

So, the height the ladder reaches is 24 feet.

b.≈ 73.74°

To determine the angle the ladder creates with the ground, use the given lengths:

Ladder distance from building

= 7 feet (adjacent to θ)

Ladder length

= 25 feet (hypotenuse)

Use cosine to determine the angle:

cos θ =

SOHCAHTOA

cos θ =

Substitute given values

θ = cos−1()

Take cos-1 of both sides

θ ≈ 73.74°

Use your calculator

3.

Recall that the cosecant is the reciprocal of the sine function, so sin θ = 3/4. One down, four more to go. Construct a right triangle to see what’s going on:

Since you know the opposite side, 3, and the hypotenuse, 4, use the Pythagorean theorem to determine the third side: (Note: This is not a 3-4-5 triangle! The hypotenuse is 4).

a2 + b2 = c2

Pythagorean theorem

(3)2 + b2 = (4)2

Substitution with a = 3 and c = 4.

 9 + b2 = 16

Perform arithmetic.

  b2 = 7

Subtract 9 from both sides.

  b =

Take the square root of both sides.

So, now that the base of the triangle is known, use it to evaluate the remaining functions:

4.Given the right triangle, use the two lengths to find the length of the third side:

a2 + b2 = c2

Pythagorean theorem

(20)2 + b2 = (52)2

Substitute a = 20 and c = 52.

400 + b2 = 2704

Simplify.

  b2 = 2304

Subtract 400 from both sides.

  b = 48

Take the square root of both sides.

Note: 20:48:52 is the fourth multiple of 5:12:13.

Now, use the side lengths to prove the statements:

a. sin θ × sec θ = tan θ

Use SOHCAHTOA and the determined side lengths.

sin = opp/hyp, sec = hyp/adj, tan = opp/adj

Simplify.

b. sin2θ + cos2θ = 1

Use SOHCAHTOA and the determined side lengths.

sin = opp/hyp, cos = adj/hyp

Simplify.

5.Construct an image to get an idea of what is going on:

In a right triangle, the two acute angles always add up to 90°. So if one angle is θ, the other acute angle is always equal to 90 − θ.

The two triangles display the relationship of each side to the respective angle in question. Pay attention to the different labeling with the different angles in question. The triangles are identical, but the sides referred to as “opposite” and “adjacent” are different, depending on which angle you’re referring to.

Now, label the angles in the same triangle and label the sides as a, b, and c:

In this triangle, evaluate the required trig functions:

sin (90 − θ) = cos θ

Given

SOHCAHTOA

Substitute the correct sides.

You could have also used a right triangle with numerical side lengths to prove this relationship.

6.Construct an image to get an idea of what is going on:

In a right triangle, the two acute angles always add up to 90°. So if one angle is θ, the other acute angle is always equal to 90 − θ.

The two triangles display the relationship of each side to the respective angle in question. Pay attention to the different labeling with the different angles in question. The triangles are identical, but the sides referred to as “opposite” and “adjacent” are different, depending on which angle you’re referring to.

Now, label the angles in the same triangle and label the sides as a, b, and c:

In this triangle, evaluate the required trig functions:

tan (90 − θ) =

Given

SOHCAHTOA

Substitute the correct sides.

Simplify.

You could have also used a right triangle with numerical side lengths to prove this relationship.

7.Given two side lengths and one angle of a triangle, use the Law of Sines:

Law of Sines

Plug in the given values using a = 7, c = 5, and A = 43 to determine the exact value first:

Substitute given values.

5 × = sin C

Multiply both sides by 5.

sin−1(5 × ) = C

Take the inverse sine of both sides.

This is the exact value as it does not have any rounding or approximation. Use a calculator (in degree mode!) to find the value of C and round to 2 decimal places:

sin−1(5 × ) = C

Exact value

29.15288537… = C

Use your calculator.

29.15 ≈ C

Approximate to 2 decimal places.

8.Given three side lengths of a triangle, use the Law of Cosines to determine angle measures. Since all three angles must be determined, use the side lengths interchangeably to determine the angles:

a2 = b2 + c2 − 2bc cos A

b2 = a2 + c2 − 2ac cos B

The Law of Cosines in all forms

c2 = a2 + b2 − 2ab cos C

Substitute the given values into each form to determine the angles:

(8)2 = (14)2 + (19)2 − 2(14)(19) cos A

Substitute the given values.

64 = 196 + 361 − 532 cos A

Simplify.

−493 = −532 cos A

Subtract (196 + 361) from both sides.

cos A

Divide both sides by −532.

cos−1() = A

Take the inverse cosine of both sides.

22.08 ≈ A

Calculator approximation of angle measure.

Repeat the process to determine angle B:

(14)2 = (8)2 + (19)2 − 2(8)(19) cos B

Substitute the given values.

196 = 64 + 361 − 304 cos B

Simplify.

−229 = −304 cos B

Subtract (64 + 361) from both sides.

= cos B

Divide both sides by −304.

cos-1() = B

Take the inverse cosine of both sides.

41.12 ≈ B

Calculator approximation of angle measure.

Now, the Law of Cosines could be used one more time in the exact same process to yield the other angle, but remember that there are 180° in a triangle and two of the three angles are known. Subtract the known angle values to find the third!

C + 22.08 + 41.12 = 180

Sum of angles in a triangle equal 180°.

C + 63.2 = 180

Perform arithmetic.

C = 116.8

Subtract 63.2 from both sides.

So, the three angle measures are approximately 22.08°, 41.12°, and 116.8°.

REFLECT

Congratulations on completing Chapter 4!

Here’s what we just covered.

Rate your confidence in your ability to:


•Understand and apply the basic trigonometric functions (sine, cosine, and tangent) and their reciprocals (secant, cosecant, and cotangent)

•Use trigonometric functions and their inverses on your calculator

•Find sine, cosine, and tangent values for complementary angles

•Use trigonometric functions to solve problems with right triangles

•Use the Pythagorean theorem to derive and understand additional trigonometric identities

•Use the Law of Sines and the Law of Cosines to solve problems with non-right triangles

If you rated any of these topics lower than you’d like, consider reviewing the corresponding lesson before moving on, especially if you found yourself unable to correctly answer one of the related end-of-chapter questions.

Access your online student tools for a handy, printable list of Key Points for this chapter. These can be helpful for retaining what you’ve learned as you continue to explore these topics.