Archive for the ‘Trigonometry’ Category

Problem 2047

Complete the sentence below.

Triangles for which two sides and the angle opposite one of them are known (SSA) are referred to as the___________________.

 

Solution:-

 

Triangles for which two sides and the angle opposite one of them are known (SSA) are referred to as the

ambiguous case.

 

Problem 2046

Determine if the statement below is true or false.

The law of sine can be used to solve triangles where three sides are known.

 

Solution:-

 

False.

 

Problem 2045

Complete the sentence below.

For a triangle with sides a, b, c, and opposite angles A, B, C, the law of sines states that__________________.

 

Solution:-

 

For a triangle with sides a, b, c, and opposite angles A, B, C, the law of sines states that

\frac{sinA}{a}=\frac{sinB}{b}=\frac{sinC}{c}

Problem 1111

Find the central angle θ which forms a sector of area 21 square feet of a circle of radius 11 feet.

 

Solution:-

 

A central angle is an angle whose vertex is at the center of a circle. The rays of a central angle subtend (intersect) an arc on the circle. The part of the circle between the rays of the angle and the arc subtended is called a sector.

Area of a sector

The area A of sector of a circle of radius r formed by a central angle of θ radians is equal to the following

A = \frac{1}{2} r2 θ

Solving for θ given θ = \frac{2A}{r^{2}}.

Substitute the values for A and r, and simplify.

θ =\frac{2A}{r^{2}}

=\frac{2 \cdot 21}{11}

\approx 0.347

Therefore, the central angle θ which forms a sector of area 21 square feet of a circle of radius 11 feet is θ = 0.347 radians.

 

 

 

Problem 1110

Find the area A of the sector of a radius 30 feet formed by the central angle \frac{1}{13} radian.

 

Solution:-

 

A central angle is an angle whose vertex is at the center of a circle. The rays of a central angle subtend (intersect) an arc on the circle. The part of the circle between the rays of the angle and the arc subtended is called a sector. We have the following theorem.

Area of a sector

The area A of sector of a circle of radius r formed by a central angle of θ radians is

A = \frac{1}{2} r2 θ

The value of r in the formula is 30 feet.

The value of θ in the formula is \frac{1}{13} radian.

Therefore,

A (area) = \frac{1}{2}*(30 feet)2 * \frac{1}{13}

= 34.615 feet2

 

 

Problem 1109

Find the length s of the arc of a circle of radius 85 centimeters subtended by the central angle 36°.

 

Solution:-

 

A central angle is an angle whose vertex is at the center of a circle. The rays of a central angle subtend (intersect) an arc on the circle. We have the following theorem.

Arc Length

For a circle of radius r, a central angle θ radians subtends an arc whose length s is

s = r θ

Convert angle in degrees to radians.

1° = \frac{\pi}{180} radian

36° = 36 *\frac{\pi}{180} radian

\approx \frac{\pi}{5}radian

s(arc length) = rθ

= 85 centimeters* \frac{\pi}{5}

=17π centimeters

\approx 53.407 centimeters

 

 

Problem 1108

Find the central angle θ which subtends an arc of length 53 miles of a circle of radius 51 miles.

 

Solution:-

 

A central angle is an angle whose vertex is at the center of a circle. The rays of a central of a circle. The rays of a central angle subtend (intersect) an arc on the circle. The arc length for a circle of radius r and a central angle of θ radians is given by s = r θ.

Solving this expression for θ given θ = \frac{s}{r}. Substitute the values for s and r.

θ = \frac{s}{r}

=\frac{53}{51}

\approx 1.039 radians

Therefore, the central angle which subtends an arc of length 53 miles of a circle of radius 51 miles is θ = 1.039 radians.

 

 

Problem 1107

Find the length s of the arc of a circle of radius 75 meters subtended by the central angle \frac{1}{25} radian.

 

Solution:-

 

A central angle is an angle whose vertex is at the center of a circle. The rays of a central angle subtend (intersect) an arc on the circle. We have the following theorem.

Arc Length

For a circle of radius r, a central angle of θ radians subtends an arc whose length s is

s = r θ

the value of r in the formula is 75 meters.

The value of θ in the formula is \frac{1}{25} radian.

s(arc length) = 75 meters * {1}{15} radian

= 3 meters

 

 

Problem 1106

Convert the angle in radians to degrees. Express your answer in decimal form.

5.63

 

Solution:-

 

Consider a circle of radius r. A central angle of a revolution (360°) will subtend an arc equal to the circumference of the circle. Because the circumference of the circle equal 2πr, we see that 360° = 2π radians, or 180° = π radians. Dividing both sides by π, we have the following conversion formula.

Converting form radians to degrees can be summarized as follows.

1 radian = \frac{180}{\pi} degrees

Therefore, the following is true.

5.63 radians = 5.63 * 1 radian

= 5.63 *\frac{180}{\pi} degrees

\approx 322.58°

 

 

Problem 1105

Convert the angle in degrees to radians.

-19°

Solution:-

 

Consider a circle of radius r. A central angle of a revolution (360°) will subtend an arc equal to the circumference of the circle. Because the circumference of the circle equal 2πr, we see that 360° = 2π radians, or 180° = π radians.

Converting form degrees to radians can be summarized as follows.

1 degree = \frac{\pi}{180} radian

Now convert the given angle in degrees to radians. Use the formula from above to write 1 degree in radians.

-19° = -19 * 1 degree

= -19 * \frac{\pi}{180} radian

Simplify. Remember that π \approx 3.1416.

-19° = -19*\frac{\pi}{180}radian

\approx-19*\frac{3.1416}{180} radian

\approx – 0.33 radian