24.1 Trigonometry Concept Overview
TRIGONOMETRY | CONCEPT OVERVIEW The topic of TRIGONOMETRY can be referenced on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
CONCEPT INTRO: TRIGONOMETRY is the study of the lengths and angles of triangles. ANGLES can be expressed in DEGREES or RADIANS. A radian is the angle that is subtended by an arc on a circle, which is equal to the radius of the circle. A DEGREE is defined as the measure of the central angle subtended by an arc of a !
circle equal to !"# the circumference of the circle. The circumference of a circle is 2𝜋𝑟 and is equal to 360° or 2𝜋 radians. One radian is approximately 57.3°. The FORMULA to CONVERT BETWEEN RADIANS AND DEGREES can be referenced under the topic of CONVERSION FACTORS on page 2 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
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It is possible to convert between radians and degrees using the following conversions: Radians = Degrees x
Degrees = Radians x
! !"#°
!"#° !
The INTERIOR ANGLES OF A POLYGON are measured in radians or degrees. The SUM of all the INTERIOR ANGLES of a polygon with n number of sides is: 𝑛 − 2 180° or (𝑛 − 2)𝜋 An ACUTE ANGLE is an angle whose measure is less than 90° 𝑜𝑟
! !
radians.
An OBTUSE ANGLE is an angle whose measure is between 90° and 180°, or ! !
and 𝜋 radians.
A RIGHT ANGLE is an angle whose measure is 90°, or
! !
radians.
The FORMULA known as the PYTHAGOREAN THEOREM is not provided in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We must memorize this formula and understand its application independent of the resources we are given come exam day. The PYTHAGOREAN THEOREM states that the squares of the lengths of a right triangle (sides a and b) are equal to the square of the length of the hypotenuse. Made with by Prepineer | Prepineer.com
Graphically illustrating this, we have:
There are a few special case right triangles that have vetted geometrical relationships that we can turn to when we are looking to solve for sides and angles. The special case triangles are known as the 30-60-90 triangle and the 45-45-90 triangle. In a 30-60-90 degree right triangle, the length of the hypotenuse is twice the length of the shorter leg, and the length of the longer leg is 3 times the length of the shorter leg. To illustrate this, a 30-60-90 triangle has the following general characteristics:
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In a 45-45-90 degree right triangle, the length of the legs are the same and the length of the hypotenuse is 2 times the length of the legs. To illustrate this, a 45-45-90 triangle has the following general characteristics:
Using the rules of these special right triangles allows us to take limited information and solve for the remaining unknown characteristics.
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TRIGONOMETRY | CONCEPT EXAMPLE The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material.
The longest side of a 30-60-90 degree right triangle is 8 3, the length of the hypotenuse is most close to: A. 2 B. 8 C. 16 D. 32
SOLUTION: The FORMULA for the PYTHAGOREAN THEOREM is not provided in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Therefore, it is a topic area that we must memorize and understand fully its application independent of the NCEES Supplied Reference Handbook. Given a 30-60-90 right triangle, we are able to refer to our special case characteristics of this geometry. In a 30-60-90 degree right triangle, the length of the hypotenuse is twice the length of the shorter leg, and the length of the longer leg is 3 times the length of the shorter leg.
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by Prepineer | Prepineer.com
To illustrate this, a 30-60-90 triangle has the following general characteristics:
We are told that the longer side is 8 3, which makes the shorter side equal to 8 and the hypotenuse is equal to 8(2) = 16. The correct answer choice is C. 𝟏𝟔
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by Prepineer | Prepineer.com
CONCEPT INTRO: TRIGONOMETRY is the study of the lengths and angles of triangles. ANGLES can be expressed in DEGREES or RADIANS. A radian is the angle that is subtended by an arc on a circle, which is equal to the radius of the circle. A DEGREE is defined as the measure of the central angle subtended by an arc of a !
circle equal to !"# the circumference of the circle. The circumference of a circle is 2𝜋𝑟 and is equal to 360° or 2𝜋 radians. One radian is approximately 57.3°. The FORMULA to CONVERT BETWEEN RADIANS AND DEGREES can be referenced under the topic of CONVERSION FACTORS on page 2 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing.
Made with
by Prepineer | Prepineer.com
It is possible to convert between radians and degrees using the following conversions: Radians = Degrees x
Degrees = Radians x
! !"#°
!"#° !
The INTERIOR ANGLES OF A POLYGON are measured in radians or degrees. The SUM of all the INTERIOR ANGLES of a polygon with n number of sides is: 𝑛 − 2 180° or (𝑛 − 2)𝜋 An ACUTE ANGLE is an angle whose measure is less than 90° 𝑜𝑟
! !
radians.
An OBTUSE ANGLE is an angle whose measure is between 90° and 180°, or ! !
and 𝜋 radians.
A RIGHT ANGLE is an angle whose measure is 90°, or
! !
radians.
The FORMULA known as the PYTHAGOREAN THEOREM is not provided in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. We must memorize this formula and understand its application independent of the resources we are given come exam day. The PYTHAGOREAN THEOREM states that the squares of the lengths of a right triangle (sides a and b) are equal to the square of the length of the hypotenuse. Made with by Prepineer | Prepineer.com
Graphically illustrating this, we have:
There are a few special case right triangles that have vetted geometrical relationships that we can turn to when we are looking to solve for sides and angles. The special case triangles are known as the 30-60-90 triangle and the 45-45-90 triangle. In a 30-60-90 degree right triangle, the length of the hypotenuse is twice the length of the shorter leg, and the length of the longer leg is 3 times the length of the shorter leg. To illustrate this, a 30-60-90 triangle has the following general characteristics:
Made with
by Prepineer | Prepineer.com
In a 45-45-90 degree right triangle, the length of the legs are the same and the length of the hypotenuse is 2 times the length of the legs. To illustrate this, a 45-45-90 triangle has the following general characteristics:
Using the rules of these special right triangles allows us to take limited information and solve for the remaining unknown characteristics.
Made with
by Prepineer | Prepineer.com
TRIGONOMETRY | CONCEPT EXAMPLE The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material.
The longest side of a 30-60-90 degree right triangle is 8 3, the length of the hypotenuse is most close to: A. 2 B. 8 C. 16 D. 32
SOLUTION: The FORMULA for the PYTHAGOREAN THEOREM is not provided in the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. Therefore, it is a topic area that we must memorize and understand fully its application independent of the NCEES Supplied Reference Handbook. Given a 30-60-90 right triangle, we are able to refer to our special case characteristics of this geometry. In a 30-60-90 degree right triangle, the length of the hypotenuse is twice the length of the shorter leg, and the length of the longer leg is 3 times the length of the shorter leg.
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by Prepineer | Prepineer.com
To illustrate this, a 30-60-90 triangle has the following general characteristics:
We are told that the longer side is 8 3, which makes the shorter side equal to 8 and the hypotenuse is equal to 8(2) = 16. The correct answer choice is C. 𝟏𝟔
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by Prepineer | Prepineer.com
