26.1 Trigonometric Ratios Concept Overview
TRIGONOMETRIC RATIOS | 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: A TRIGONOMETRIC RATIO relates the sides of a triangle to a specific geometric angle within it using three basic trigonometric functions. Knowing how to apply such relationships allows us to use trigonometric ratios to solve for the vertical and horizontal components of a force, as well as the magnitude of the force, amongst other things. Given a right triangle, with a combination of sides and angles defined, we can use TRIGONOMETRIC RATIOS to determine any other unknowns that relate to that particular triangle. The driving FORMULAS for TRIGONOMETRIC RATIOS can be referenced under the SUBJECT of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. There are three basic trigonometric functions: SINE, COSINE, and TANGENT.
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Even without the reference of what is outlined for us in the NCEES Reference Handbook, we are able to develop of the cuff each TRIGONOMETRIC RATIO using a standard schematic of a RIGHT TRIANGLE such that:
Given a right triangle, you can define the SINE, COSINE, and TANGENT of either of the non-90Β° angles, such athat: SINE:
sin π =
πππππ ππ‘π π¦ = π»π¦πππ‘πππ’π π π
COSINE:
cos π =
π΄πππππππ‘ π₯ = π»π¦πππ‘πππ’π π π
TANGENT:
tan π =
πππππ ππ‘π π¦ = π΄πππππππ‘ π₯ Made with
by Prepineer | Prepineer.com
Another hack in remembering these TRIGONOMETRIC RATIOS quickly off hand is by recalling a story that many of us may have heard in High School, and that is of the great Indian Chief SOH CAH TOA. SOH CAH TOA is an ACRONYM that we personally revert to any time we are dealing with a TRIGONOMETRIC RATIO. Itβs easy to recall and quickly gives us the template for each relationship. SOH CAH TOA visually lays out like this: β’ (S)ine THETA is equal to (O)pposite over (H)ypotenuse β’ (C)osine THETA is equal to (A)djacent over (H)ypotenuse β’ (T)angent THETA is equal to (O)pposite over (A)djacent So taking each 3 letter ACRONYM, we are able to develop the appropriate TRIGONOMETRIC RATIO. The FORMULAS highlighting the RECIPROCAL TRIGONOMETRIC RATIOS can be referenced under the topic of TRIGONOMETRY on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. There are three additional TRIGONOMETRIC RATIOS called the RECIPROCAL TRIGONOMETRIC RATIOS. These RATIONS are not typically used to solve for unknowns within a given 90o triangle, but are often used in higher-level mathematics applications.
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The RECIPROCAL TRIGONOMETRIC RATIOS are defined as: COSECANT:
csc π =
1 π»π¦πππ‘πππ’π π = sin π πππππ ππ‘π
SECANT:
sec π =
1 π»π¦πππ‘πππ’π π = cos π π΄πππππππ‘
COTANGENT:
cot π =
1 π΄πππππππ‘ = tan π πππππ ππ‘π
NOTE: It is important to know that the sinβ1 , cos β1 , and tanβ1 key on your calculator do not represent the COSECANT, SECANT, and COTANGENT function, but rather the inverse SINE, COSINE, and TANGENT which represents a completely different function. The sign of the trigonometric function varies depending on which QUADRANT we are working, based on the signs of βπ₯β and βπ¦β.
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by Prepineer | Prepineer.com
Graphically illustrating this:
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TRIGONOMETRIC RATIOS | CONCEPT EXAMPLE The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material. 3
Given that sin π = 5 and that the angle π is acute, cos π is best represented as: A. cos π =
2 5 4
B. cos π = 5 4
C. cos π = β 5 4
D. cos π = 5
SOLUTION: The FORMULAS for all TRIGONOMETRIC RATIOS can be referenced under the topic of TRIGONOMETRY on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. In this problem, we are given a defined TRIGONOMETRIC RATIO and asked what another RATIO would be based on what is given and knowing the reference angle is ACUTE. Remember that we can either flip through the NCEES Reference Handbook for these RATIOS, or quickly off hand recall them using a story that many of us may have heard in High School, and that is of the great Indian Chief SOH CAH TOA. Made with
by Prepineer | Prepineer.com
SOH CAH TOA is an ACRONYM that we personally revert to any time we are dealing with a TRIGONOMETRIC RATIO. Itβs easy to recall and quickly gives us the template for each relationship. SOH CAH TOA visually lays out like this: β’ (S)ine THETA is equal to (O)pposite over (H)ypotenuse β’ (C)osine THETA is equal to (A)djacent over (H)ypotenuse β’ (T)angent THETA is equal to (O)pposite over (A)djacent Taking each 3 letter ACRONYM, we are able to develop the appropriate TRIGONOMETRIC RATIO we are asked for. In this problem, we are given:
sin π =
3 5
The SOH of SOH CAH TOA tells us that: β’ (S)ine THETA is equal to (O)pposite over (H)ypotenuse This establishes that: β’ OPPOSITE SIDE (y): 3 β’ HYPOTENUSE (h): 5
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by Prepineer | Prepineer.com
Knowing this data, we need to define one more side to know the length of all sides in this particular triangle. We will deploy our knowledge of the PYTHAGOREAN THEOREM. The GENERAL FORMULA of 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 NCEES Supplied Reference Handbook. The Pythagorean Theorem tells us: π2 + π 2 = π 2 Where a and b are the sides of a triangle, and c is the hypotenuse of that same triangle. Plugging in the values we know, we get: π2 + 32 = 52 Rearranging and solving for our unknown side, βπβ, we find: π = Β±4
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If the angle π is ACUTE, then π resides in QUANDRANT I, where cos π > 0, visually illustrating this:
So all of the COMPONENT VALUES of our RATIO will be positive. Referring back to SOH CAH TOA, lets remember that for COH: β’ (C)osine THETA is equal to (A)djacent over (H)ypotenuse With all of our sides defined, we have: β’ OPPOSITE SIDE (O): 3 β’ HYPOTENUSE (H): 5 β’ ADJACENT SIDE (A): 4
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Which gives us:
cos π =
π΄πππππππ‘ π₯ 4 = = π»π¦πππ‘πππ’π π π 5 π
The correct answer choice is B. ππ¨π¬ π = π
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TRIGONOMETRIC RATIOS | CONCEPT EXAMPLE 3
Given that sin π = 5 and that the angle π is obtuse, cos π is best represented as: 3
A. cos π = β 5 4
B. cos π = β 5 C. cos π =
1 5 4
D. cos π = 5
SOLUTION: The FORMULAS for all TRIGONOMETRIC RATIOS can be referenced under the topic of TRIGONOMETRY on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. In this problem, we are given a defined TRIGONOMETRIC RATIO and asked what another RATIO would be based on what is given and knowing the reference angle is OBTUSE. Remember that we can either flip through the NCEES Reference Handbook for these RATIOS, or quickly off hand recall them using a story that many of us may have heard in High School, and that is of the great Indian Chief SOH CAH TOA.
Made with
by Prepineer | Prepineer.com
SOH CAH TOA is an ACRONYM that we personally revert to any time we are dealing with a TRIGONOMETRIC RATIO. Itβs easy to recall and quickly gives us the template for each relationship. SOH CAH TOA visually lays out like this: β’ (S)ine THETA is equal to (O)pposite over (H)ypotenuse β’ (C)osine THETA is equal to (A)djacent over (H)ypotenuse β’ (T)angent THETA is equal to (O)pposite over (A)djacent Taking each 3 letter ACRONYM, we are able to develop the appropriate TRIGONOMETRIC RATIO we are asked for. In this problem, we are given:
sin π =
3 5
The SOH of SOH CAH TOA tells us that: β’ (S)ine THETA is equal to (O)pposite over (H)ypotenuse This establishes that: β’ OPPOSITE SIDE (y): 3 β’ HYPOTENUSE (h): 5
Made with
by Prepineer | Prepineer.com
Knowing this data, we need to define one more side to know the length of all sides in this particular triangle. We will deploy our knowledge of the PYTHAGOREAN THEOREM. The GENERAL FORMULA of 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 NCEES Supplied Reference Handbook. The Pythagorean Theorem tells us: π2 + π 2 = π 2 Where a and b are the sides of a triangle, and c is the hypotenuse of that same triangle. Plugging in the values we know, we get: π2 + 32 = 52 Rearranging and solving for our unknown side, βπβ, we find: π = Β±4
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If the angle π is OBTUSE, then π resides in QUANDRANT II, where cos π < 0, visually illustrating this:
So give us an x COMPONENT VALUE (ADJACENT) that is NEGATIVE and a y COMPONENT VALUE (OPPOSITE) that is POSITIVE. Referring back to SOH CAH TOA, lets remember that for COH: β’ (C)osine THETA is equal to (A)djacent over (H)ypotenuse With all of our sides defined, we have: β’ OPPOSITE SIDE (O): 3 β’ HYPOTENUSE (H): 5 β’ ADJACENT SIDE (A): -4
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Which gives us:
cos π =
π΄πππππππ‘ π₯ 4 = =β π»π¦πππ‘πππ’π π π 5 π
The correct answer choice is B. ππ¨π¬ π = β π
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by Prepineer | Prepineer.com
CONCEPT INTRO: A TRIGONOMETRIC RATIO relates the sides of a triangle to a specific geometric angle within it using three basic trigonometric functions. Knowing how to apply such relationships allows us to use trigonometric ratios to solve for the vertical and horizontal components of a force, as well as the magnitude of the force, amongst other things. Given a right triangle, with a combination of sides and angles defined, we can use TRIGONOMETRIC RATIOS to determine any other unknowns that relate to that particular triangle. The driving FORMULAS for TRIGONOMETRIC RATIOS can be referenced under the SUBJECT of MATHEMATICS on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. There are three basic trigonometric functions: SINE, COSINE, and TANGENT.
Made with
by Prepineer | Prepineer.com
Even without the reference of what is outlined for us in the NCEES Reference Handbook, we are able to develop of the cuff each TRIGONOMETRIC RATIO using a standard schematic of a RIGHT TRIANGLE such that:
Given a right triangle, you can define the SINE, COSINE, and TANGENT of either of the non-90Β° angles, such athat: SINE:
sin π =
πππππ ππ‘π π¦ = π»π¦πππ‘πππ’π π π
COSINE:
cos π =
π΄πππππππ‘ π₯ = π»π¦πππ‘πππ’π π π
TANGENT:
tan π =
πππππ ππ‘π π¦ = π΄πππππππ‘ π₯ Made with
by Prepineer | Prepineer.com
Another hack in remembering these TRIGONOMETRIC RATIOS quickly off hand is by recalling a story that many of us may have heard in High School, and that is of the great Indian Chief SOH CAH TOA. SOH CAH TOA is an ACRONYM that we personally revert to any time we are dealing with a TRIGONOMETRIC RATIO. Itβs easy to recall and quickly gives us the template for each relationship. SOH CAH TOA visually lays out like this: β’ (S)ine THETA is equal to (O)pposite over (H)ypotenuse β’ (C)osine THETA is equal to (A)djacent over (H)ypotenuse β’ (T)angent THETA is equal to (O)pposite over (A)djacent So taking each 3 letter ACRONYM, we are able to develop the appropriate TRIGONOMETRIC RATIO. The FORMULAS highlighting the RECIPROCAL TRIGONOMETRIC RATIOS can be referenced under the topic of TRIGONOMETRY on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. There are three additional TRIGONOMETRIC RATIOS called the RECIPROCAL TRIGONOMETRIC RATIOS. These RATIONS are not typically used to solve for unknowns within a given 90o triangle, but are often used in higher-level mathematics applications.
Made with
by Prepineer | Prepineer.com
The RECIPROCAL TRIGONOMETRIC RATIOS are defined as: COSECANT:
csc π =
1 π»π¦πππ‘πππ’π π = sin π πππππ ππ‘π
SECANT:
sec π =
1 π»π¦πππ‘πππ’π π = cos π π΄πππππππ‘
COTANGENT:
cot π =
1 π΄πππππππ‘ = tan π πππππ ππ‘π
NOTE: It is important to know that the sinβ1 , cos β1 , and tanβ1 key on your calculator do not represent the COSECANT, SECANT, and COTANGENT function, but rather the inverse SINE, COSINE, and TANGENT which represents a completely different function. The sign of the trigonometric function varies depending on which QUADRANT we are working, based on the signs of βπ₯β and βπ¦β.
Made with
by Prepineer | Prepineer.com
Graphically illustrating this:
Made with
by Prepineer | Prepineer.com
TRIGONOMETRIC RATIOS | CONCEPT EXAMPLE The following problem introduces the concept reviewed within this module. Use this content as a primer for the subsequent material. 3
Given that sin π = 5 and that the angle π is acute, cos π is best represented as: A. cos π =
2 5 4
B. cos π = 5 4
C. cos π = β 5 4
D. cos π = 5
SOLUTION: The FORMULAS for all TRIGONOMETRIC RATIOS can be referenced under the topic of TRIGONOMETRY on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. In this problem, we are given a defined TRIGONOMETRIC RATIO and asked what another RATIO would be based on what is given and knowing the reference angle is ACUTE. Remember that we can either flip through the NCEES Reference Handbook for these RATIOS, or quickly off hand recall them using a story that many of us may have heard in High School, and that is of the great Indian Chief SOH CAH TOA. Made with
by Prepineer | Prepineer.com
SOH CAH TOA is an ACRONYM that we personally revert to any time we are dealing with a TRIGONOMETRIC RATIO. Itβs easy to recall and quickly gives us the template for each relationship. SOH CAH TOA visually lays out like this: β’ (S)ine THETA is equal to (O)pposite over (H)ypotenuse β’ (C)osine THETA is equal to (A)djacent over (H)ypotenuse β’ (T)angent THETA is equal to (O)pposite over (A)djacent Taking each 3 letter ACRONYM, we are able to develop the appropriate TRIGONOMETRIC RATIO we are asked for. In this problem, we are given:
sin π =
3 5
The SOH of SOH CAH TOA tells us that: β’ (S)ine THETA is equal to (O)pposite over (H)ypotenuse This establishes that: β’ OPPOSITE SIDE (y): 3 β’ HYPOTENUSE (h): 5
Made with
by Prepineer | Prepineer.com
Knowing this data, we need to define one more side to know the length of all sides in this particular triangle. We will deploy our knowledge of the PYTHAGOREAN THEOREM. The GENERAL FORMULA of 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 NCEES Supplied Reference Handbook. The Pythagorean Theorem tells us: π2 + π 2 = π 2 Where a and b are the sides of a triangle, and c is the hypotenuse of that same triangle. Plugging in the values we know, we get: π2 + 32 = 52 Rearranging and solving for our unknown side, βπβ, we find: π = Β±4
Made with
by Prepineer | Prepineer.com
If the angle π is ACUTE, then π resides in QUANDRANT I, where cos π > 0, visually illustrating this:
So all of the COMPONENT VALUES of our RATIO will be positive. Referring back to SOH CAH TOA, lets remember that for COH: β’ (C)osine THETA is equal to (A)djacent over (H)ypotenuse With all of our sides defined, we have: β’ OPPOSITE SIDE (O): 3 β’ HYPOTENUSE (H): 5 β’ ADJACENT SIDE (A): 4
Made with
by Prepineer | Prepineer.com
Which gives us:
cos π =
π΄πππππππ‘ π₯ 4 = = π»π¦πππ‘πππ’π π π 5 π
The correct answer choice is B. ππ¨π¬ π = π
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by Prepineer | Prepineer.com
TRIGONOMETRIC RATIOS | CONCEPT EXAMPLE 3
Given that sin π = 5 and that the angle π is obtuse, cos π is best represented as: 3
A. cos π = β 5 4
B. cos π = β 5 C. cos π =
1 5 4
D. cos π = 5
SOLUTION: The FORMULAS for all TRIGONOMETRIC RATIOS can be referenced under the topic of TRIGONOMETRY on page 23 of the NCEES Supplied Reference Handbook, Version 9.4 for Computer Based Testing. In this problem, we are given a defined TRIGONOMETRIC RATIO and asked what another RATIO would be based on what is given and knowing the reference angle is OBTUSE. Remember that we can either flip through the NCEES Reference Handbook for these RATIOS, or quickly off hand recall them using a story that many of us may have heard in High School, and that is of the great Indian Chief SOH CAH TOA.
Made with
by Prepineer | Prepineer.com
SOH CAH TOA is an ACRONYM that we personally revert to any time we are dealing with a TRIGONOMETRIC RATIO. Itβs easy to recall and quickly gives us the template for each relationship. SOH CAH TOA visually lays out like this: β’ (S)ine THETA is equal to (O)pposite over (H)ypotenuse β’ (C)osine THETA is equal to (A)djacent over (H)ypotenuse β’ (T)angent THETA is equal to (O)pposite over (A)djacent Taking each 3 letter ACRONYM, we are able to develop the appropriate TRIGONOMETRIC RATIO we are asked for. In this problem, we are given:
sin π =
3 5
The SOH of SOH CAH TOA tells us that: β’ (S)ine THETA is equal to (O)pposite over (H)ypotenuse This establishes that: β’ OPPOSITE SIDE (y): 3 β’ HYPOTENUSE (h): 5
Made with
by Prepineer | Prepineer.com
Knowing this data, we need to define one more side to know the length of all sides in this particular triangle. We will deploy our knowledge of the PYTHAGOREAN THEOREM. The GENERAL FORMULA of 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 NCEES Supplied Reference Handbook. The Pythagorean Theorem tells us: π2 + π 2 = π 2 Where a and b are the sides of a triangle, and c is the hypotenuse of that same triangle. Plugging in the values we know, we get: π2 + 32 = 52 Rearranging and solving for our unknown side, βπβ, we find: π = Β±4
Made with
by Prepineer | Prepineer.com
If the angle π is OBTUSE, then π resides in QUANDRANT II, where cos π < 0, visually illustrating this:
So give us an x COMPONENT VALUE (ADJACENT) that is NEGATIVE and a y COMPONENT VALUE (OPPOSITE) that is POSITIVE. Referring back to SOH CAH TOA, lets remember that for COH: β’ (C)osine THETA is equal to (A)djacent over (H)ypotenuse With all of our sides defined, we have: β’ OPPOSITE SIDE (O): 3 β’ HYPOTENUSE (H): 5 β’ ADJACENT SIDE (A): -4
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by Prepineer | Prepineer.com
Which gives us:
cos π =
π΄πππππππ‘ π₯ 4 = =β π»π¦πππ‘πππ’π π π 5 π
The correct answer choice is B. ππ¨π¬ π = β π
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