# Trigonometric equations and angle between vectors.

Question #1:

You used Pythagorus' theorem to determine whether or not a triangle was a right triangle. The sides of the triangle are:

a = sqrt(416), b = sqrt(601), and c = sqrt(1009) so that a2 + b2 did not equal c2. Thus it is not a right triangle. Let the α, β, γ be the angles of the triangle across from sides a, b, c respectively.

Use the law of cosines to determine how far the angle γ across from c deviates from 90 degrees. (Round all answers to 2 decimal places.)

1. Use the law of sines to determine the other 2 angles α and β of the triangle.

2. Does (α + β + γ) = 180 degrees? (yes or no)

3. Find the area of the triangle using theorem 11.4, p903.

4. Find the area of the triangle using Heron's formula (theorem 11.6, p914).

5. Do your answers in (4) and (5) agree? (yes or no)

6. If your answer is no, which answer do you think is correct? The one from (4) or the one from (5)? Both of them are wrong?

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#### Solution Preview

Question #1:

You used Pythagorus' theorem to determine whether or not a triangle was a right triangle. The sides of the triangle are:

a = sqrt(416), b = sqrt(601), and c = sqrt(1009) so that a2 + b2 did not equal c2. Thus it is not a right triangle. Let the α, β, γ be the angles of the triangle across from sides a, b, c respectively.

Use the law of cosines to determine how far the angle γ across from c deviates from 90 degrees. (Round all answers to 2 decimal places.)

1. Use the law of sines to determine the other 2 angles α and β of the triangle.

Solution:

2. Does (α + β + γ) = 180 degrees? (yes or no)

Solution:

α + β + γ = 39.95°+50.51°+89.54° = 180°

Yes, sum is 180 degrees.

3. Find the area of the triangle using theorem 11.4, p903.

Solution:

Area = 250 square units

4. Find the area of the triangle using Heron's formula (theorem 11.6, p914).

Solution:

a = √416 = 20.396, b = √601 = 24.515, c = √1009 = 31.76

s = (a+b+c)/2 = (20.396+24.515+31.76)/2 = 38.3355

Area

≈ 250 square units

5. Do your answers in (4) and (5) agree? (yes or no)

Solution:

Yes, both the answers are the same.

6. If your answer is no, which answer do you think is correct? The one from (4) or the one from (5)? Both of them are wrong?

Question #2:

A vector is like an arrow with a length and a direction. A vector as measured from the origin has it tail at (0,0) and its tip at (x,y).

= (x,y) = x + y

= |P|( cos (θP) + sin(θP) )

where |P| = sqrt(x2 + y2), cos(θP) = x/|P|, sin(θP) = y/|P|.

Let = (3,4) and = (5,-12).

1. Plot (x,y) for on graph paper

Solution:

2. Draw the vector on your graph paper.

3. What is |P|?

Solution:

|P| =

4. What is θP?

Solution:

cos(θP) = x/|P|= 3/5 = 0.6

θP = 53.13°

5. Draw θP on your graph.

Solution:

6. What reference angle αP corresponds to θP?

Solution:

αP = 53.13°

7. Plot (x,y) for .

Solution:

8. Draw the vector on your graph paper

9. What is |Q|?

Solution:

|Q| =

10. What is θQ?

Solution:

sin(θQ) = y/|Q|= -12/13

θQ = 292.62°

11. Draw θQ on your graph.

Solution:

12. What reference angle αQ corresponds to θQ?

Solution:

αQ = 360°-292.62° = 67.38°

The vector PQ is defined as = - = + (- ), where (- ) = (-x,-y).

13. What are the components of ?

Solution:

PQ = (5-3, -12-4) = (2,-16)

14. Using the vector addition laws (p. 1014), draw on your graph starting by placing its tail on the tip of , so as to obtain + ...

#### Solution Summary

This posting includes step by step solutions to some questions on pendulum, graphing trigonometric functions, solving trigonometric equations and angle between vectors.