Math  /  Trigonometry

QuestionE.g. Solve the following triangles: a. b. P=1809037=53cos37119qtan37=r19q=19cos37r=19tan37q23.8r14.3\begin{array}{l} \angle P=180^{\circ}-90^{\circ}-37^{\circ}=53^{\circ} \\ \cos 37^{\circ}-1 \frac{19}{q} \\ \tan 37^{\circ}=\frac{r}{19} \\ q=\frac{19}{\cos 37^{\circ}} \\ r=19 \tan 37^{\circ} \\ q \doteq 23.8 \\ r \doteq 14.3 \end{array} P=53,q23.8 cm,r19.3 cm\therefore \angle P=53^{\circ}, q \doteq 23.8 \mathrm{~cm}, r \equiv 19.3 \mathrm{~cm}

Studdy Solution

STEP 1

1. We are working with a right triangle, as indicated by the presence of a 9090^\circ angle.
2. The given angle 3737^\circ is one of the non-right angles in the triangle.
3. We need to find the remaining angle and the lengths of the sides opposite and adjacent to the given angle.

STEP 2

1. Calculate the missing angle in the triangle.
2. Use trigonometric identities to solve for the unknown side lengths.
3. Calculate the approximate values of the side lengths.

STEP 3

Calculate the missing angle P \angle P in the triangle using the sum of angles in a triangle, which is 180180^\circ:
P=1809037 \angle P = 180^\circ - 90^\circ - 37^\circ P=53 \angle P = 53^\circ

STEP 4

Use the cosine function to find the length of the side adjacent to the 3737^\circ angle, denoted as q q . The cosine of an angle in a right triangle is the ratio of the adjacent side to the hypotenuse:
cos37=19q \cos 37^\circ = \frac{19}{q}
Rearrange to solve for q q :
q=19cos37 q = \frac{19}{\cos 37^\circ}

STEP 5

Use the tangent function to find the length of the side opposite the 3737^\circ angle, denoted as r r . The tangent of an angle in a right triangle is the ratio of the opposite side to the adjacent side:
tan37=r19 \tan 37^\circ = \frac{r}{19}
Rearrange to solve for r r :
r=19tan37 r = 19 \tan 37^\circ

STEP 6

Calculate the approximate values of q q and r r using a calculator:
q23.8 q \approx 23.8 r14.3 r \approx 14.3
The solution for the triangle is:
P=53,q23.8cm,r14.3cm \angle P = 53^\circ, \, q \approx 23.8 \, \text{cm}, \, r \approx 14.3 \, \text{cm}

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