Question 1

Solve the problem.
-$\csc ^ { 2 } 65 ^ { \circ } - \tan ^ { 2 } 25 ^ { \circ }$

Accepted Answer

The expression simplifies using the trigonometric identity $\csc^2 \theta - \tan^2 \theta = 1$, where $\theta = 65^\circ$ and $90^\circ - \theta = 25^\circ$.

Question 2

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given
angle. Give exact answers with rational denominators.
-Find $\cos \mathrm { A }$ when $\mathrm { a } = \sqrt { 13 }$ and $\mathrm { c } = 10$.

Accepted Answer

Given that side $a = \sqrt{13}$ and hypotenuse $c = 10$, we can find side $b$ using the Pythagorean theorem: $a^2 + b^2 = c^2$. Substituting the given values, we get $\sqrt{13}^2 + b^2 = 10^2$, which simplifies to $13 + b^2 = 100$. Solving for $b^2$, we find $b^2 = 87$, so $b = \sqrt{87}$. The cosine of angle A, $\cos A$, is adjacent over hypotenuse, which is $\frac{\sqrt{87}}{10}$.

Question 3

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given
angle. Give exact answers with rational denominators.
-Find sin B when b = 4 and c = 7.

Accepted Answer

A

Question 4

Solve the problem.
-$\cos 35 ^ { \circ } \sin 55 ^ { \circ } + \sin 35 ^ { \circ } \cos 55 ^ { \circ }$

Accepted Answer

This expression can be simplified using the sine addition formula: $\sin(A + B) = \sin A \cos B + \cos A \sin B$. Here, $A = 35^\circ$ and $B = 55^\circ$, so the expression simplifies to $\sin(35^\circ + 55^\circ) = \sin(90^\circ)$, which equals 1.

Question 5

Solve the right triangle using the information given. Round answers to two decimal places, if necessary. 
-$a = 5 , \alpha = 25 ^ { \circ }$; Find $b , c$, and $\beta$.

Accepted Answer

The answer of Solve the right triangle using the information...

Question 6

Solve the right triangle using the information given. Round answers to two decimal places, if necessary. 
-$\mathrm { a } = 3 , \mathrm { c } = 8 ;$ Find $\mathrm { b } , \alpha$, and $\beta$.

Accepted Answer

Given $a = 3$ and $c = 8$, we can use the Pythagorean theorem ($a^2 + b^2 = c^2$) to find $b$. Solving for $b$, we get $b = \sqrt{c^2 - a^2} = \sqrt{8^2 - 3^2} = \sqrt{64 - 9} = \sqrt{55} \approx 7.42$. To find the angles, we can use trigonometric ratios. Since $a$ and $b$ are known, we can use the sine, cosine, or tangent functions. For example, $\sin(\alpha) = \frac{a}{c} = \frac{3}{8}$, which gives $\alpha \approx 22.02^\circ$, and $\beta = 90^\circ - \alpha \approx 67.98^\circ$, assuming this is a right-angled triangle. Therefore, the correct answer is $b = 7.42$, $\alpha = 22.02^\circ$, and $\beta = 67.98^\circ$, which matches option B.

Question 7

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given
angle. Give exact answers with rational denominators.
-Find $\csc B$ when $a = 9$ and $b = 7$.

Accepted Answer

The csc of an angle is the reciprocal of the sine of that angle. In a right triangle, $\sin B = \frac{opposite}{hypotenuse}$. Given $a = 9$ and $b = 7$, we first find the hypotenuse $c$ using Pythagoras' theorem: $c = \sqrt{a^2 + b^2} = \sqrt{9^2 + 7^2} = \sqrt{81 + 49} = \sqrt{130}$. Thus, $\sin B = \frac{a}{c} = \frac{9}{\sqrt{130}}$, and $\csc B = \frac{1}{\sin B} = \frac{\sqrt{130}}{9}$. However, since $B$ is adjacent to $b$, and $c$ is the hypotenuse, the correct setup should be $\sin B = \frac{b}{c} = \frac{7}{\sqrt{130}}$, leading to $\csc B = \frac{1}{\sin B} = \frac{\sqrt{130}}{7}$, which matches choice A.

Question 8

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given
angle. Give exact answers with rational denominators.
-Find $\cot \mathrm { A }$ when $b = 5$ and $c = 6$.

Accepted Answer

The answer of Two sides of a right triangle ABC...

Question 9

Solve the problem.
-Find $\tan P$.

Accepted Answer

The answer of Solve the problem.
-Find $\tan P$.

...

Question 10

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given
angle. Give exact answers with rational denominators.
-Find $\cot A$ when $a = 3$ and $c = 4$.

Accepted Answer

The answer of Two sides of a right triangle ABC...

Question 11

Solve the problem.
-A surveyor is measuring the distance across a small lake. He has set up his transit on one side of the lake 90 feet from a piling that is directly across from a pier on the other side of the lake. From his transit, the angle between
The piling and the pier is 55°. What is the distance between the piling and the pier to the nearest foot?

Accepted Answer

The answer of Solve the problem.
-A surveyor is measuring the...

Question 12

Solve the right triangle using the information given. Round answers to two decimal places, if necessary. 
-$a = 7 , b = 4 ;$ Find $c , \alpha$, and $\beta$.

Accepted Answer

The answer of Solve the right triangle using the information...

Question 13

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given
angle. Give exact answers with rational denominators.
-Find cos A when a = 3 and b = 7.

Accepted Answer

The answer of Two sides of a right triangle ABC...

Question 14

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given
angle. Give exact answers with rational denominators.
-Find sec $B$ when $a = 9$ and $b = 5$.

Accepted Answer

To find sec(B), we need the hypotenuse, which is $\sqrt{a^2 + b^2} = \sqrt{9^2 + 5^2} = \sqrt{81 + 25} = \sqrt{106}$. Sec(B) is the reciprocal of cos(B), which is adjacent/hypotenuse = 9/$\sqrt{106}$. Rationalizing gives $\frac{\sqrt{106}}{9}$.

Question 15

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given
angle. Give exact answers with rational denominators.
-Find sin A when a = 8 and b = 7.

Accepted Answer

The answer of Two sides of a right triangle ABC...

Question 16

Solve the right triangle using the information given. Round answers to two decimal places, if necessary. 
-$a = 5 , \beta = 35 ^ { \circ }$; Find $b , c$, and $\alpha$.

Accepted Answer

The answer of Solve the right triangle using the information...

Question 17

Solve the problem.
-$\tan 15 ^ { \circ } - \frac { \cos 75 ^ { \circ } } { \cos 15 ^ { \circ } }$

Accepted Answer

Using trigonometric identities, $\tan 15^\circ = 2 - \sqrt{3}$ and $\cos 75^\circ = \sin 15^\circ = \frac{\sqrt{6} - \sqrt{2}}{4}$, $\cos 15^\circ = \frac{\sqrt{6} + \sqrt{2}}{4}$. Simplifying $\frac{\cos 75^\circ}{\cos 15^\circ}$ gives $\tan 15^\circ$, so $\tan 15^\circ - \frac{\cos 75^\circ}{\cos 15^\circ} = 0$.

Question 18

Solve the problem.
-A radio transmission tower is 210 feet tall. How long should a guy wire be if it is to be attached 13 feet from the top and is to make an angle of 20° with the ground? Give your answer to the nearest tenth of a foot.

Accepted Answer

The answer of Solve the problem.
-A radio transmission tower is...

Question 19

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given
angle. Give exact answers with rational denominators.
-Find $\tan B$ when $a = 4$ and $b = 5$.

Accepted Answer

In a right triangle, $\tan B = \frac{opposite}{adjacent}$, so $\tan B = \frac{a}{b} = \frac{4}{5}$. However, since $a = 4$ and $b = 5$ are given for sides opposite and adjacent to angle B respectively, the correct calculation should be $\tan B = \frac{b}{a} = \frac{5}{4}$, making option D correct.

Question 20

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given
angle. Give exact answers with rational denominators.
-Find $\tan \mathrm { A }$ when $\mathrm { a } = 2$ and $\mathrm { b } = 9$.

Accepted Answer

In a right triangle, $\tan A = \frac{\text{opposite}}{\text{adjacent}}$. Here, side $a = 2$ is opposite angle A, and side $b = 9$ is adjacent to angle A, so $\tan A = \frac{2}{9}$.

Quiz 7: Applications of Trigonometric Functions

Solve the problem. - $\csc ^ { 2 } 65 ^ { \circ } - \tan ^ { 2 } 25 ^ { \circ }$

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find $\cos \mathrm { A }$ when $\mathrm { a } = \sqrt { 13 }$ and $\mathrm { c } = 10$ .

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find sin B when b = 4 and c = 7.

Solve the problem. - $\cos 35 ^ { \circ } \sin 55 ^ { \circ } + \sin 35 ^ { \circ } \cos 55 ^ { \circ }$

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find $\csc B$ when $a = 9$ and $b = 7$ .

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find $\cot \mathrm { A }$ when $b = 5$ and $c = 6$ .

Solve the problem. -Find $\tan P$ . $Solve the problem. -Find \tan P . A) \tan P = \frac { 3 } { 5 } B) \tan P = \frac { 4 } { 3 } C) \tan \mathrm { P } = \frac { 3 } { 4 } D) \tan P = \frac { 4 } { 5 }$

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find $\cot A$ when $a = 3$ and $c = 4$ .

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find cos A when a = 3 and b = 7.

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find sec $B$ when $a = 9$ and $b = 5$ .

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find sin A when a = 8 and b = 7.

Solve the problem. - $\tan 15 ^ { \circ } - \frac { \cos 75 ^ { \circ } } { \cos 15 ^ { \circ } }$

Solve the problem. -A radio transmission tower is 210 feet tall. How long should a guy wire be if it is to be attached 13 feet from the top and is to make an angle of 20° with the ground? Give your answer to the nearest tenth of a foot.

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find $\tan B$ when $a = 4$ and $b = 5$ .

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find $\tan \mathrm { A }$ when $\mathrm { a } = 2$ and $\mathrm { b } = 9$ .

Quiz 7: Applications of Trigonometric Functions

Solve the problem. - csc⁡265∘−tan⁡225∘\csc ^ { 2 } 65 ^ { \circ } - \tan ^ { 2 } 25 ^ { \circ }csc265∘−tan225∘

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find cos⁡A\cos \mathrm { A }cosA when a=13\mathrm { a } = \sqrt { 13 }a=13​ and c=10\mathrm { c } = 10c=10 .

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find sin B when b = 4 and c = 7.

Solve the problem. - cos⁡35∘sin⁡55∘+sin⁡35∘cos⁡55∘\cos 35 ^ { \circ } \sin 55 ^ { \circ } + \sin 35 ^ { \circ } \cos 55 ^ { \circ }cos35∘sin55∘+sin35∘cos55∘

Solve the right triangle using the information given. Round answers to two decimal places, if necessary. - a=5,α=25∘a = 5 , \alpha = 25 ^ { \circ }a=5,α=25∘ ; Find b,cb , cb,c , and β\betaβ .

Solve the right triangle using the information given. Round answers to two decimal places, if necessary. - a=3,c=8;\mathrm { a } = 3 , \mathrm { c } = 8 ;a=3,c=8; Find b,α\mathrm { b } , \alphab,α , and β\betaβ .

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find csc⁡B\csc BcscB when a=9a = 9a=9 and b=7b = 7b=7 .

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find cot⁡A\cot \mathrm { A }cotA when b=5b = 5b=5 and c=6c = 6c=6 .

Solve the problem. -Find tan⁡P\tan PtanP .

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find cot⁡A\cot AcotA when a=3a = 3a=3 and c=4c = 4c=4 .

Solve the right triangle using the information given. Round answers to two decimal places, if necessary. - a=7,b=4;a = 7 , b = 4 ;a=7,b=4; Find c,αc , \alphac,α , and β\betaβ .

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find cos A when a = 3 and b = 7.

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find sec BBB when a=9a = 9a=9 and b=5b = 5b=5 .

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find sin A when a = 8 and b = 7.

Solve the right triangle using the information given. Round answers to two decimal places, if necessary. - a=5,β=35∘a = 5 , \beta = 35 ^ { \circ }a=5,β=35∘ ; Find b,cb , cb,c , and α\alphaα .

Solve the problem. - tan⁡15∘−cos⁡75∘cos⁡15∘\tan 15 ^ { \circ } - \frac { \cos 75 ^ { \circ } } { \cos 15 ^ { \circ } }tan15∘−cos15∘cos75∘​

Solve the problem. -A radio transmission tower is 210 feet tall. How long should a guy wire be if it is to be attached 13 feet from the top and is to make an angle of 20° with the ground? Give your answer to the nearest tenth of a foot.

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find tan⁡B\tan BtanB when a=4a = 4a=4 and b=5b = 5b=5 .

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find tan⁡A\tan \mathrm { A }tanA when a=2\mathrm { a } = 2a=2 and b=9\mathrm { b } = 9b=9 .

Solve the problem. - $\csc ^ { 2 } 65 ^ { \circ } - \tan ^ { 2 } 25 ^ { \circ }$

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find $\cos \mathrm { A }$ when $\mathrm { a } = \sqrt { 13 }$ and $\mathrm { c } = 10$ .

Solve the problem. - $\cos 35 ^ { \circ } \sin 55 ^ { \circ } + \sin 35 ^ { \circ } \cos 55 ^ { \circ }$

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find $\csc B$ when $a = 9$ and $b = 7$ .

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find $\cot \mathrm { A }$ when $b = 5$ and $c = 6$ .

Solve the problem. -Find $\tan P$ . $Solve the problem. -Find \tan P . A) \tan P = \frac { 3 } { 5 } B) \tan P = \frac { 4 } { 3 } C) \tan \mathrm { P } = \frac { 3 } { 4 } D) \tan P = \frac { 4 } { 5 }$

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find $\cot A$ when $a = 3$ and $c = 4$ .

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find sec $B$ when $a = 9$ and $b = 5$ .

Solve the problem. - $\tan 15 ^ { \circ } - \frac { \cos 75 ^ { \circ } } { \cos 15 ^ { \circ } }$

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find $\tan B$ when $a = 4$ and $b = 5$ .

Two sides of a right triangle ABC (C is the right angle) are given. Find the indicated trigonometric function of the given angle. Give exact answers with rational denominators. -Find $\tan \mathrm { A }$ when $\mathrm { a } = 2$ and $\mathrm { b } = 9$ .