Solve Trigonometry Angles: Degrees, Minutes & Seconds

[Drain Brain]

please help me to solve these problems,

1. how many degrees, minutes, and seconds are respectively passed over in $11\frac{1}{9}$ minutes by the hour and minute hands of a watch?

2. The number of degrees in one acute angle of a right-angled triangle is equal to the number of grades in the other; express both angles in degrees.

 

[mathmari]

2. The number of degrees in one acute angle of a right-angled triangle is equal to the number of grades in the other; express both angles in degrees.

View attachment 2986

The sum of the angles of a triangle is equal to $180^{\circ}$.

Therefore:

$$\hat{A}+\hat{B}+\hat{C}=180^{\circ} \Rightarrow 90^{\circ}+\hat{B}+\hat{C}=180^{\circ} \Rightarrow \hat{B}+\hat{C}=90^{\circ}$$

"The number of degrees in one acute angle of a right-angled triangle is equal to the number of grades in the other":
The two acute angles of the above right-angles trianle are $\hat{B}$ and $\hat{C}$.

So, $\hat{B}=\hat{C}$.

Replacing this at the equation $\hat{B}+\hat{C}=90^{\circ}$, we get the following:

$$\hat{B}+\hat{C}=90^{\circ} \Rightarrow 2 \hat{B}=90^{\circ} \Rightarrow \hat{B}=\hat{C}=45^{\circ}$$

 

[MarkFL]

please help me to solve these problems,

1. how many degrees, minutes, and seconds are respectively passed over in $11\frac{1}{9}$ minutes by the hour and minute hands of a watch?

2. The number of degrees in one acute angle of a right-angled triangle is equal to the number of grades in the other; express both angles in degrees.

Can you show us what you have tried? This way our helpers know where you are stuck and can better help.

View attachment 2986

The sum of the angles of a triangle is equal to $180^{\circ}$.

Therefore:

$$\hat{A}+\hat{B}+\hat{C}=180^{\circ} \Rightarrow 90^{\circ}+\hat{B}+\hat{C}=180^{\circ} \Rightarrow \hat{B}+\hat{C}=90^{\circ}$$

"The number of degrees in one acute angle of a right-angled triangle is equal to the number of grades in the other":
The two acute angles of the above right-angles trianle are $\hat{B}$ and $\hat{C}$.

So, $\hat{B}=\hat{C}$.

Replacing this at the equation $\hat{B}+\hat{C}=90^{\circ}$, we get the following:

$$\hat{B}+\hat{C}=90^{\circ} \Rightarrow 2 \hat{B}=90^{\circ} \Rightarrow \hat{B}=\hat{C}=45^{\circ}$$

I believe the OP was saying that one angle is measured in gradians while the other it in degrees. 400 gradians is equivalent to 360 degrees.

 

[mathmari]

I believe the OP was saying that one angle is measured in gradians while the other it in degrees. 400 gradians is equivalent to 360 degrees.

Oh, I'm sorry.. (Lipssealed)(Wasntme)

The angle $A$ is $90^{\circ}$.
Let the acute angle $B$ be measured in gradians, so the angle $B$ is $x$ gradians which is equal to $\frac{360 x}{400}=0.9 x$ degrees.
The angle $C$ is $y$ degrees.

"The number of degrees in one acute angle of a right-angled triangle is equal to the number of grades in the other": $x=y$

The sum of the angles of the triangle is equal to $180$ degrees.

$$90^{\circ}+0.9 x+y=180^{\circ} \Rightarrow 0.9 x+x=90^{\circ} \Rightarrow 1.9x=90^{\circ} \Rightarrow x=\left (\frac{90}{1.9}\right )^{\circ}$$

Therefore, the angle $B$ is $ \left (0.9\cdot \frac{90}{1.9}\right )^{\circ}$ and the angle $C$ is $\left (\frac{90}{1.9}\right )^{\circ}$.

 

[CellCoree]

1. To solve this problem, we first need to understand the relationship between degrees, minutes, and seconds. One degree is equal to 60 minutes, and one minute is equal to 60 seconds. Therefore, to convert minutes to degrees, we divide by 60, and to convert seconds to degrees, we divide by 3600.

For the hour hand, it travels 360 degrees in 12 hours, so in 1 minute, it will travel $\frac{360}{12\times60} = \frac{1}{2}$ degrees. In 11 $\frac{1}{9}$ minutes, it will travel $11\frac{1}{9} \times \frac{1}{2} = \frac{11}{18}$ degrees. This can also be expressed as 11 degrees and 10 minutes.

For the minute hand, it travels 360 degrees in 60 minutes, so in 1 minute, it will travel 6 degrees. In 11 $\frac{1}{9}$ minutes, it will travel $11\frac{1}{9} \times 6 = 66\frac{2}{9}$ degrees. This can also be expressed as 66 degrees and 13 minutes.

Therefore, in total, the hour hand will have passed over 11 degrees and 10 minutes, while the minute hand will have passed over 66 degrees and 13 minutes.

2. Let us represent the two acute angles of a right-angled triangle as $\theta$ and $\alpha$. Since we know that the sum of all angles in a triangle is 180 degrees, we can write the following equation:

$\theta + \alpha + 90 = 180$

Solving for $\alpha$, we get:

$\alpha = 180 - \theta - 90 = 90 - \theta$

Since we are told that the number of degrees in one acute angle is equal to the number of grades in the other, we can write the following equation:

$\theta = \frac{\alpha}{100}$

Substituting this into our previous equation, we get:

$\frac{\alpha}{100} + \alpha + 90 = 180$

Solving for $\alpha$, we get:

$\alpha = \frac{10}{11} \times 90 = 81\frac{9}{11}$

Therefore, one angle is 81 degrees and 9 minutes, while the other angle is 8