Sam’s 8 Tips for Trigonometry (4103) 416 Math
1) Calculator must always be on DEG mode
2) SIN COS TAN & PYTHAGORAS are for right angle triangles only
3) SIN COS TAN are always attached to ANGLES
4) Pythagoras = a2 + b2 = c2 (“c” has to be the hypotenuse / longest side)
5) ALL triangles have 180° / ALL corners are 90° / ALL flat lines are 180°
6) When “x” is on the bottom, switch it
7) When “#” is on the bottom, multiply it on the other side
8) Will always press 2nd FUNCTION when determining the value of an angle
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Here is an easy way to remember the relationships for trig functions and the right triangle.
SOH - CAH - TOA
It is pronounced "so - ka - toe - ah".
The SOH stands for "Sine of an angle is Opposite over Hypotenuse."
The CAH stands for "Cosine of an angle is Adjacent over Hypotenuse."
The TOA stands for "Tangent of an angle is Opposite over Adjacent."
There are several ways to understand why a certain input angle produces a certain output value. At first, the most important manner of understanding this is tied to right triangles. All of the trigonometric values for angles between 0 degrees and 90 degrees can be understood by considering this diagram:
1) Calculator must always be on DEG mode
2) SIN COS TAN & PYTHAGORAS are for right angle triangles only
3) SIN COS TAN are always attached to ANGLES
4) Pythagoras = a2 + b2 = c2 (“c” has to be the hypotenuse / longest side)
5) ALL triangles have 180° / ALL corners are 90° / ALL flat lines are 180°
6) When “x” is on the bottom, switch it
7) When “#” is on the bottom, multiply it on the other side
8) Will always press 2nd FUNCTION when determining the value of an angle
___________________________________________________________________________________________________________________________________
Here is an easy way to remember the relationships for trig functions and the right triangle.
SOH - CAH - TOA
It is pronounced "so - ka - toe - ah".
The SOH stands for "Sine of an angle is Opposite over Hypotenuse."
The CAH stands for "Cosine of an angle is Adjacent over Hypotenuse."
The TOA stands for "Tangent of an angle is Opposite over Adjacent."
There are several ways to understand why a certain input angle produces a certain output value. At first, the most important manner of understanding this is tied to right triangles. All of the trigonometric values for angles between 0 degrees and 90 degrees can be understood by considering this diagram:
Note: The "hypothenuse" is always the longest side AND opposite the right (90 degree) angle. The "opposite" and "adjacent" sides vary, depnding on the angle we are working with.
The names of the three primary trigonometry functions are:
sine / cosine / tangent
These are abbreviated this way:
sine.....sin
cosine.....cos
tangent.....tan
We will be concerned with angle A. Notice that the sides of the triangle are labeled appropriately "opposite side" and "adjacent side" relative to angle A. The hypotenuse is not considered opposite or adjacent to the angle A.
We will also be concerned with length of the three sides. For this discussion we will call the "length of the opposite side" simply the "opposite". Similarly, the other two lengths will be called "adjacent" and "hypotenuse".
The value for the sine of angle A is defined as the value that you get when you divide the opposite side by the hypotenuse. This can be written:
sin(A) = opposite / hypotenuse
Or simply:
sin(A) = opp / hyp
Or, even more simplified:
sin(A) = o / h
Suppose we measure the lengths of the sides of this triangle. Here are some realistic values:
We will also be concerned with length of the three sides. For this discussion we will call the "length of the opposite side" simply the "opposite". Similarly, the other two lengths will be called "adjacent" and "hypotenuse".
The value for the sine of angle A is defined as the value that you get when you divide the opposite side by the hypotenuse. This can be written:
sin(A) = opposite / hypotenuse
Or simply:
sin(A) = opp / hyp
Or, even more simplified:
sin(A) = o / h
Suppose we measure the lengths of the sides of this triangle. Here are some realistic values:
This would mean that:
sin(A) = opposite / hypotenuse = 4.00 cm / 7.21 cm = 0.5548
Or simply:
sin(A) = 0.5548
Now for the other two trig functions.
The value for the cosine of angle A is defined as the value that you get when you divide the adjacent side by the hypotenuse. This can be written:
cos(A) = adjacent / hypotenuse;
cos(A) = adj / hyp (or)
cos(A) = a / h
Using the above measured triangle, this would mean that:
cos(A) = adjacent / hypotenuse = 6.00 cm / 7.21 cm = 0.8322
Or simply:
cos(A) = 0.8322
The value for the tangent of angle A is defined as the value that you get when you divide the opposite side by the adjacent side. This can be written:
tan(A) = opposite / adjacent;
tan(A) = opp / adj (or)
tan(A) = o / a
Using the above measured triangle, this would mean that:
tan(A) = opposite / adjacent = 4.00 cm / 6.00 cm = 0.6667
Or simply:
tan(A) = 0.6667
The angle A in the above triangle is actually very close to 33.7 degrees. So, we would say:
0.5548 = sin(33.7°)
0.8322 = cos(33.7°)
0.6667 = tan(33.7°)
When we are dealing with Oblique triangles (any triangle that is not a right triangle), we can no longer use SOH CAH TOA, rather we have to choose between SINE LAW or COSINE LAW.
How do we choose between the 2 laws?
Ask yourself the following question:
Do I have a side/angle pair??
*Reminder: “BIG LETTERS” (A,B,C) represent angles and “small letters” (a,b,c) represent sides.
This means: Do I have both values of; A & a / B & b (or) C & c
*Only one pair is needed in order to answer "YES" to that question.
If you answer YES = SINELAW
If you answer NO = COSINE LAW
Formulas:
Sinelaw: a = b = c
SinA SinB SinC
Cosine Law: a² = b² + c² - [2bcCosA] (or)
b² = a² + c² - [2acCosB] (or)
c² = a² + b² - [2abCosC]
sin(A) = opposite / hypotenuse = 4.00 cm / 7.21 cm = 0.5548
Or simply:
sin(A) = 0.5548
Now for the other two trig functions.
The value for the cosine of angle A is defined as the value that you get when you divide the adjacent side by the hypotenuse. This can be written:
cos(A) = adjacent / hypotenuse;
cos(A) = adj / hyp (or)
cos(A) = a / h
Using the above measured triangle, this would mean that:
cos(A) = adjacent / hypotenuse = 6.00 cm / 7.21 cm = 0.8322
Or simply:
cos(A) = 0.8322
The value for the tangent of angle A is defined as the value that you get when you divide the opposite side by the adjacent side. This can be written:
tan(A) = opposite / adjacent;
tan(A) = opp / adj (or)
tan(A) = o / a
Using the above measured triangle, this would mean that:
tan(A) = opposite / adjacent = 4.00 cm / 6.00 cm = 0.6667
Or simply:
tan(A) = 0.6667
The angle A in the above triangle is actually very close to 33.7 degrees. So, we would say:
0.5548 = sin(33.7°)
0.8322 = cos(33.7°)
0.6667 = tan(33.7°)
When we are dealing with Oblique triangles (any triangle that is not a right triangle), we can no longer use SOH CAH TOA, rather we have to choose between SINE LAW or COSINE LAW.
How do we choose between the 2 laws?
Ask yourself the following question:
Do I have a side/angle pair??
*Reminder: “BIG LETTERS” (A,B,C) represent angles and “small letters” (a,b,c) represent sides.
This means: Do I have both values of; A & a / B & b (or) C & c
*Only one pair is needed in order to answer "YES" to that question.
If you answer YES = SINELAW
If you answer NO = COSINE LAW
Formulas:
Sinelaw: a = b = c
SinA SinB SinC
Cosine Law: a² = b² + c² - [2bcCosA] (or)
b² = a² + c² - [2acCosB] (or)
c² = a² + b² - [2abCosC]