Proof of the trigonometric ratios of complementary allied angles
Two acute angles are complementary to each other if their sum is equal to 90°. In a right triangle the sum of the two acute angles is equal to 90°. So, the two acute angles of a right triangle are always complementary to each other.
Let ABC be a right triangle, right angled at B
If <ACB = θ, then <BAC = 90° – θ and hence the angles <BAC and <ACB are complementary
For the angle θ, we have
Similarly, for the angle (90° – θ), we have
Comparing the equations in (1) and (2) we get,
Trigonometric Ratios of Complementary Angles
Examples: Evaluate : cos 56° / sin 34°
The angles 56° and 34° are complementary.
So, using trigonometric ratios of complementary angles, we have
cos 56° = cos (90° – 56°) = sin 34°
cos 56° / sin 34° = sin 34° / sin 34° = 1
Hence the value of cos 56° / sin 34° is 1.
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In geometry, an arc is a part of the circumference of a circle or sphere. The length of an arc is measured in radians or degrees. A central angle is an angle whose vertex is at the center of a circle and whose sides are chords of the circle. The measure of a central angle is equal to the arc it intercepts. A chord is a line segment that joins two points on a circle. An inscribed angle is an angle whose vertex is on a circle and whose sides are chords of the circle. The measure of an inscribed angle is equal to half the measure of the arc it intercepts. A semicircle is a part of a circle that is bounded by a diameter and an arc. The measure of a semicircle is 180 degrees. A tangent is a line that intersects a circle at exactly one point. The point of intersection is called the point of tangency. A tangent to a circle is perpendicular to the radius at the point of tangency.
The relation between arc radius and angle is as follows:
The length of an arc is equal to the product of the radius of the circle and the measure of the arc in radians.
The measure of a central angle is equal to the arc it intercepts.
The measure of an inscribed angle is equal to half the measure of the arc it intercepts.
The measure of a semicircle is 180 degrees.
A tangent to a circle is perpendicular to the radius at the point of tangency.
Here are some examples of how the relation between arc radius and angle can be used in real life:
When you are driving a car, you use the arc radius and angle to determine how far you need to turn the steering wheel in order to make a turn.
When you are building a house, you use the arc radius and angle to determine how long the rafters need to be in order to support the roof.
When you are designing a bridge, you use the arc radius and angle to determine how long the cables need to be in order to support the weight of the bridge.
The relation between arc radius and angle is a fundamental concept in geometry. It is used in many different applications, from driving a car to building a house.
What is a circle?
A circle is a round shape with all sides of equal length and all angles of equal measure.
What is a radius?
The radius of a circle is the distance from the center of the circle to any point on the edge of the circle.
What is a diameter?
The diameter of a circle is the distance across the circle, through the center. The diameter is twice the radius.
What is an arc?
An arc is a part of a circle. It is bounded by two endpoints and is not the entire circle.
What is a central angle?
A central angle is an angle that has its vertex at the center of a circle. The measure of a central angle is equal to the measure of the arc it intercepts.
What is an inscribed angle?
An inscribed angle is an angle that has its vertex on the circumference of a circle. The measure of an inscribed angle is equal to half the measure of the arc it intercepts.
What is a tangent?
A tangent is a line that intersects a circle at exactly one point. The point of intersection is called the point of tangency.
What is a secant?
A secant is a line that intersects a circle in two points.
What is a chord?
A chord is a line segment that joins two points on a circle.
What is a major arc?
A major arc is an arc that is greater than a semicircle.
What is a minor arc?
A minor arc is an arc that is less than a semicircle.
What is a sector?
A sector is a part of a circle that is bounded by two radii and an arc.
What is a segment?
A segment is a part of a circle that is bounded by two chords.
A pie slice is a sector of a circle that is often used to represent a part of a whole.
What is the difference between a circle and a sphere?
A circle is a two-dimensional shape, while a sphere is a three-dimensional shape. A circle is the set of all points in a plane that are the same distance from a given point, called the center. A sphere is the set of all points in space that are the same distance from a given point, called the center.
What is the difference between a radius and a diameter?
The radius of a circle is the distance from the center of the circle to any point on the edge of the circle. The diameter of a circle is the distance across the circle, through the center. The diameter is twice the radius.
What is the difference between an arc and a chord?
An arc is a part of a circle. It is bounded by two endpoints and is not the entire circle. A chord is a line segment that joins two points on a circle.
What is the difference between a central angle and an inscribed angle?
A central angle is an angle that has its vertex at the center of a circle. The measure of a central angle is equal to the measure of the arc it intercepts. An inscribed angle is an angle that has its vertex on the circumference of a circle. The measure of an inscribed angle is equal to half the measure of the arc it intercepts.
What is the difference between a tangent and a secant?
A tangent is a line that intersects a circle at exactly one point. The point of intersection is called the point of tangency. A secant is a line that intersects a circle in two points.
What is the difference between a major arc and a minor arc?
A major arc is an arc that is greater than a semicircle. A minor arc is an arc that is less than a semicircle.
What is the difference between a sector and a segment?
A sector is a part of a circle that is bounded by two radii and an arc. A segment is a part of a circle that is bounded by two chords.
What is the difference between a pie slice and a sector?
A pie slice is a sector of a circle that is often used to represent a part of a whole. A sector is a part of a circle that is bounded by two radii and an arc.
Sure. Here are some MCQs without mentioning the topic Relation Between Arc Radius And Angle:
A circle is a round shape with all points the same distance from the center. What is the distance from any point on the circle to the center called?
(A) Radius
(B) Diameter
(C) Circumference
(D) Arc
A diameter is a straight line that goes through the center of a circle and has endpoints on the circle. What is the length of a diameter of a circle with a radius of 5 inches?
(A) 5 inches
(B) 10 inches
(C) 15 inches
(D) 20 inches
The circumference of a circle is the distance around the circle. What is the formula for the circumference of a circle with a radius of 5 inches?
(A) $c = 5\pi$
(B) $c = 10\pi$
(C) $c = 15\pi$
(D) $c = 20\pi$
An arc is a part of the circumference of a circle. What is the measure of an arc that is half of the circumference of a circle?
(A) $90^\circ$
(B) $180^\circ$
(C) $270^\circ$
(D) $360^\circ$
A sector is a part of a circle that is bounded by an arc and two radii. What is the area of a sector with a radius of 5 inches and an arc measure of $90^\circ$?
(A) $25\pi$ square inches
(B) $50\pi$ square inches
(C) $75\pi$ square inches
(D) $100\pi$ square inches
