Question

For △ABC, which side is opposite angle B?

side AC

side AB

side BC


Angle D in △EDF is a right angle.

What is the value of tan F ?

4/3

3/4

3/5

4/5

What is the trigonometric ratio for sin Z ?
Enter your answer, as a simplified fraction, in the boxes.

What is the trigonometric ratio for cosD ?
Enter your answer, as a simplified fraction, in the boxes.

Tessa has leaned a ladder against the side of her house. The ladder forms a 63˚ angle with the ground and rests against the house at a spot that is 8 meters high.

Which length is the best approximation for the distance along the ground from the bottom of the ladder to the wall?
2 m
3 m
4 m
5 m

Answer 1

Problem 1

Answer: side AC

---------------

Explanation:

Side AC is opposite angle B because it is the furthest you can get on the triangle to be away from point B. This segment is a leg of the right triangle. Note how the letter B is not anywhere to be found in AC.

===========================================
Problem 2

Answer: 4/3

---------------

Explanation:

angle D is the right angle, the side across from this right angle is the hypotenuse EF

with reference angle F, we have
opposite = ED = 4
adjacent = DF = 3

tan(angle) = opposite/adjacent
tan(F) = ED/DF
tan(F) = 4/3

===========================================
Problem 3

Answer: 3/5

---------------

Explanation:

Use the Pythagorean Theorem to find the missing side XY
a^2 + b^2 = c^2
(XY)^2 + (YZ)^2 = (XZ)^2
(XY)^2 + (32)^2 = (40)^2
(XY)^2 + 1024 = 1600
(XY)^2 + 1024-1024 = 1600-1024
(XY)^2 = 576
sqrt[ (XY)^2 ] = sqrt[ 576 ]
XY = 24

Use that value to find the trig ratio we need

sin(angle) = (opposite)/(hypotenuse)
sin(Z) = (XY)/(XZ)
sin(Z) = (24)/(40)
sin(Z) = (8*3)/(8*5)
sin(Z) = 3/5

===========================================
Problem 4

Answer: 7/25

---------------

Explanation:

cos(angle) = adjacent/hypotenuse
cos(D) = (ED)/(DF)
cos(D) = (21)/(75)
cos(D) = (7*3)/(25*3)
cos(D) = 7/25

===========================================
Problem 5

Answer: 4 meters

---------------

Explanation:

We want to find the horizontal distance from the ground to the foot of the ladder. Call this x for now. 

This x is the adjacent leg, since it is touching the 63 degree angle. The opposite side is 8 meters

opposite = 8
adjacent = x

Use the tangent function to solve for x

tan(angle) = opposite/adjacent
tan(63) = 8/x
x*tan(63) = x*(8/x)
x*tan(63) = 8
x*tan(63)/tan(63) = 8/tan(63)
x = 4.07620359595543
Rounding to the nearest whole number, we get 4 meters.

Answer 2

The relationships between the sides and angles of the right triangles in

the question are given by trigonometric ratios.

Correct responses;

• 1. Side AC

• $\displaystyle 2. \hspace{0.3 cm}\frac{4}{3}$

• $\displaystyle 3. \hspace{0.3 cm}\frac{3}{5}$

• $\displaystyle 4. \hspace{0.3 cm}\frac{7}{25}$

• 5. 4 m.

Methods used to arrive at the above responses;

1. The given triangle is ΔABC

The side opposite angle B is the side not bearing the letter B which is the side AC

2. The tangent of the angle, F, is given by the following formula;

$\displaystyle tan(F) = \mathbf{\frac{Opposite \ side \ to \ angle \ F}{Adjacent \ side \ to \ angle \ F}}$

Therefore;

• $\displaystyle \underline{ tan(F) = \frac{4}{3}}$

3. The sine of the angle Z is given by the following formula;

$\displaystyle sin (Z) = \mathbf{ \frac{Opposite \ side \ to \ angle \ Z}{Hypotenuse \ side \ of \ right \ triangle}}$

Therefore;

$\displaystyle sin(Z) = \mathbf{\frac{XY}{40}}$

XY = √(40² - 32²) = 24

Which gives;

• $\displaystyle \underline{sin(Z) = \frac{24}{40} = \frac{3}{5}}$

4. The trigonometric ratio for cos(D) is given as follows;

$\displaystyle cos(D) = \mathbf{ \frac{Adjacent \ side \ to \ angle \ \angle D}{Hypotenuse \ side \ of \ right \ triangle}}$

Therefore;

$\displaystyle cos(D) = \frac{21}{75} = \frac{7}{25}$

• $\displaystyle \underline{ cos(D) = \frac{7}{25} }$

5. The tangent of the angle 63° is expressed as follows;

$\displaystyle tan(63^{\circ}) = \mathbf{\frac{8 \, m}{Distance \ AB \ (ground \ distance \ from \ wall)}}$

Therefore;

$\displaystyle Distance \ AB = \frac{8 \, m}{tan(63^{\circ})} \approx \mathbf{4.076 \, m}$

• The approximate distance from the bottom of the ladder o the wall is 4 m

Learn more about trigonometric ratios here:

https://brainly.com/question/8517005