Question

For questions $1-10$ solve $\mathrm{△}\mathrm{ABC}$ given $\mathrm{AC}=29\mathrm{cm},\mathrm{BC}=19\mathrm{cm},B={49}^{\circ }$

Short answer

Based on the step-by-step solution provided, answer the following question: Question: Find the measures of the angles and the missing side of triangle ABC, given that side AC = 29 cm, side BC = 19 cm, and angle B = 49 degrees. Answer: To find the missing angle measures and side of triangle ABC, first, use the Law of Cosines to determine the length of side AB. Next, apply the Law of Sines to calculate angle A. Finally, use the angle-sum property to find angle C. After completing these steps, you will have: - Angle A: (A°) - Angle B: 49° - Angle C: (C°) - Side AC: 29 cm - Side BC: 19 cm - Side AB: (AB cm)

Step-by-step solution

Use the Law of Cosines to find the missing side

We are given sides AC and BC, as well as angle B. The Law of Cosines states that: $A{B}^2=A{C}^2+B{C}^2-2(AC)(BC)\cos B$ Plug in the given values: $A{B}^2={29}^2+{19}^2-2(29)(19)\cos {49}^{\circ }$ Calculate the value of $A{B}^2$ and then take the square root to find the length of side $AB$.

Use the Law of Sines to find angle A

Now that we have the length of side $AB$, we can use the Law of Sines to find angle A. The Law of Sines states that: $\frac{\sin A}{AC}=\frac{\sin B}{AB}$ Plug in the given values and the obtained value for side $AB$: $\frac{\sin A}{29}=\frac{\sin {49}^{\circ }}{AB}$ Solve for angle A by multiplying both sides by 29 and then taking the inverse sine of the result.

Use the angle-sum property to find angle C

The angle-sum property of a triangle states that the sum of the interior angles of a triangle is equal to 180 degrees. Therefore, we can find angle C by subtracting angles A and B from 180 degrees. $C={180}^{\circ }-A-B$ Plug in the obtained value of angle A and the given value of angle B to find angle C.

State the measures of the angles and sides of the triangle

Now that we have found all the angles and the missing side length of the triangle, we can write the measures of the angles and sides of triangle ABC: $A=(A^{\circ })$ $B={49}^{\circ }$ $C=(C^{\circ })$ $AC=29cm$ $BC=19cm$ $AB=(ABcm)$

Key concepts

Law of Cosines

The Law of Cosines is a critical concept when solving triangles in trigonometry, especially when you're dealing with sides and angles. It's an extension of the Pythagorean theorem but for any kind of triangle, not just the right-angled ones. The Law of Cosines formula is given by:
\[ c^2 = a^2 + b^2 - 2ab \text{cos}(C) \[
where $c$ is the side opposite of angle $C$, and $a$ and $b$ are the other two sides. This formula helps to find a missing side when you know two sides and the angle between them—as was the case in our example with triangle ABC where we found the length of side AB.

Understanding this concept is crucial, as it can be applied to various scenarios, such as finding the distance between two points in navigation or physics problems involving forces. Remember, when using the Law of Cosines, always ensure that the angle and its opposite side are both known or calculable quantities.

Law of Sines

Whereas the Law of Cosines related the sides to one angle, the Law of Sines provides a relationship between all sides and angles of a triangle. It's particularly useful when you have a pair of an angle and its opposite side, just like a bridge that connects two parts of a riverbank. The formula is given by:
$$\frac{\sin (A)}{a}=\frac{\sin (B)}{b}=\frac{\sin (C)}{c}$$
This equal ratio signifies that the sine of each angle over its opposite side is a constant for all three angles and sides in a triangle. In our example, after finding side AC using the Law of Cosines, we used the Law of Sines to figure out angle A. One quick tip: When using the Law of Sines, make sure to look out for the 'ambiguous case,' where two different triangles might satisfy the given conditions - this occurs when solving for angles with given sides.

Triangle Angle-Sum Property

The Triangle Angle-Sum Property is as straightforward as its name suggests. When piecing together a triangle, imagine you're creating your own three-cornered hat—every triangle is just like that hat, where the sum of its angles always totals 180 degrees. The simple formula here is:
$$A+B+C={180}^{\circ }$$
Once we know two of the angles, as we did with triangle ABC after utilizing the Law of Cosines and the Law of Sines, calculating the third angle is a cakewalk. You just subtract the sum of the known angles from 180 degrees. It's essential to use this property to ensure the internal consistency of the triangle and is often the last step to fully solving a triangle, as was demonstrated in our exercise when we found angle C.