Triangles have 3 sides and their angles add up to 180 degrees. A right triangle is a triangle with a right (90 degree) angle, quelle grande surprise. This is denoted using the perpendicular symbol (a box) at the right angle. The hypotenuse is the longest is of a right triangle and always directly opposite of the right angle.
Pythagorean Theorem
Using the Pythagorean theorem, if 2 sides of the triangle are given, the 3rd side can be found using the following equation:
a2 + b2 = c2
The key to remember is that "c" is always the hypotenuse in this equation. The other 2 sides are transitive, so either side can be called "a" or "b."
Basic Trig Functions
Trigonometry uses proportions of sides and angles in a right triangle. The key to trig is being able to label the sides correctly based on the angle that is being used.
The mantra to remember to be successful with trigonometry is:
SOH CAH TOA
sin of the angle = opposite / hypotenuse (sinθ = o/a)
cosine of the angle = adjacent /hypotenuse (cosθ = a/h)
tan of the angle = opposite / adjacent (tanθ = o/a)
For most of physics, your calculator needs to be set in degrees, not radians. Below is a chart of when to use each equation or concept:
UnknownGivensEquation1 side
other 2 sides
a2 + b2 = c2
1 angle*
other angle*
angle1 + angle2 = 90 degrees
1 side
another side and 1 angle*
Trig Functions
SOH CAH TOA
1 angle*
2 sides
Inverse Trig Functions
sin-1(o/h) cos-1(a/h) tan-1(o/a)
* angle referred to is a non-right angle
Whiteboard Worksheets has a very good online interactive applet that will let you see what happens to the length of the sides are you move the triangle. Unfortunately, it will soon ask you for a password (which I do not have), but play around with it while you can to see what can help you.
Example 1:
On the triangle below; Side a is 8 cm and Side b is 6 cm. What is the magnitude of side c?
Because we are given 2 sides and our unknown is the third side, we would have to use the Pythagorean theorem.
c = ? a = 8 cm b = 6 cm
a2 + b2 = c2
√ (a2 + b2) = c
√ (82 + 62) = c
√ (64 + 36) = c
√ (100) = c
10 cm = c
Example 2:
If on the triangle below Side a is 10 inches and Side c is 15 inches, how long is Side b?
Because we are given 2 sides and our unknown is the third side, we would have to use the Pythagorean theorem.
c = 15 inches a = 10 inches b = ?
a2 + b2 = c2
b2 = c2 - a2
b = √(c2 - a2)
b = √(152 - 102)
b = √(225 - 100)
b = √(125)
b = 11.2 inches
Example 3:
A right triangle has a non-right angle of θ and a hypotenuse of Z. Write out the equation for Side R in terms of Z and θ.
Because we are given 1 side and 1 angle, and we are looking for a second side, so we will use the trig functions to solve.
θ = θ hypotenuse (h) = Z adjacent (a) = R
cosθ = a/h
a = hcosθ
R = Zcosθ
Example 4:
The hypotenuse of a right triangle is 1875 ohms(Ω). If side in between an angle and the right angle is 1125 ohms, how big is the angle?
Because we are given 2 sides and need to find an angle, we will have to use the inverse trig functions. Remember to label your sides first
θ = ? hypotenuse (h) = 1875 ohms adjacent (a) = 1125 ohms
cosθ = a/h
θ = cos-1 (a/h)
θ = cos-1 (1125/1875)
θ = 53.13°
Example 5:
The hypotenuse of a right triangle is 1875 ohms(Ω). If side across from a non-right angle is 1500 ohms, how big is the angle?
Because we are given 2 sides and need to find an angle, we will have to use the inverse trig functions. Remember to label your sides first
θ = ? hypotenuse (h) = 1875 ohms opposite (a) = 1500 ohms
sinθ = o/h
θ = sin-1 (o/h)
θ = sin-1 (1500/1875)
θ = 53.13°
Example 6:
If side across from a non-right angle is 1500 ohms and the side between this angle and the right angle is 1125 ohms, how big is the angle?
Because we are given 2 sides and need to find an angle, we will have to use the inverse trig functions. Remember to label your sides first
θ = ? adjacent (a) = 1125 ohms opposite (a) = 1500 ohms
tanθ = o/a
θ = tan-1 (o/a)
θ = tan-1 (1500/1125)
θ = 53.13°
Pythagorean Theorem
Using the Pythagorean theorem, if 2 sides of the triangle are given, the 3rd side can be found using the following equation:
a2 + b2 = c2
The key to remember is that "c" is always the hypotenuse in this equation. The other 2 sides are transitive, so either side can be called "a" or "b."
Basic Trig Functions
Trigonometry uses proportions of sides and angles in a right triangle. The key to trig is being able to label the sides correctly based on the angle that is being used.
- The hypotenuse is always opposite of the right angle, ALWAYS
- The opposite side is on the opposite side of the triangle from the angle that is being used
- The adjacent side is between the angle used and the right angle
The mantra to remember to be successful with trigonometry is:
SOH CAH TOA
sin of the angle = opposite / hypotenuse (sinθ = o/a)
cosine of the angle = adjacent /hypotenuse (cosθ = a/h)
tan of the angle = opposite / adjacent (tanθ = o/a)
For most of physics, your calculator needs to be set in degrees, not radians. Below is a chart of when to use each equation or concept:
UnknownGivensEquation1 side
other 2 sides
a2 + b2 = c2
1 angle*
other angle*
angle1 + angle2 = 90 degrees
1 side
another side and 1 angle*
Trig Functions
SOH CAH TOA
1 angle*
2 sides
Inverse Trig Functions
sin-1(o/h) cos-1(a/h) tan-1(o/a)
* angle referred to is a non-right angle
Whiteboard Worksheets has a very good online interactive applet that will let you see what happens to the length of the sides are you move the triangle. Unfortunately, it will soon ask you for a password (which I do not have), but play around with it while you can to see what can help you.
Example 1:
On the triangle below; Side a is 8 cm and Side b is 6 cm. What is the magnitude of side c?
Because we are given 2 sides and our unknown is the third side, we would have to use the Pythagorean theorem.
c = ? a = 8 cm b = 6 cm
a2 + b2 = c2
√ (a2 + b2) = c
√ (82 + 62) = c
√ (64 + 36) = c
√ (100) = c
10 cm = c
Example 2:
If on the triangle below Side a is 10 inches and Side c is 15 inches, how long is Side b?
Because we are given 2 sides and our unknown is the third side, we would have to use the Pythagorean theorem.
c = 15 inches a = 10 inches b = ?
a2 + b2 = c2
b2 = c2 - a2
b = √(c2 - a2)
b = √(152 - 102)
b = √(225 - 100)
b = √(125)
b = 11.2 inches
Example 3:
A right triangle has a non-right angle of θ and a hypotenuse of Z. Write out the equation for Side R in terms of Z and θ.
Because we are given 1 side and 1 angle, and we are looking for a second side, so we will use the trig functions to solve.
θ = θ hypotenuse (h) = Z adjacent (a) = R
cosθ = a/h
a = hcosθ
R = Zcosθ
Example 4:
The hypotenuse of a right triangle is 1875 ohms(Ω). If side in between an angle and the right angle is 1125 ohms, how big is the angle?
Because we are given 2 sides and need to find an angle, we will have to use the inverse trig functions. Remember to label your sides first
θ = ? hypotenuse (h) = 1875 ohms adjacent (a) = 1125 ohms
cosθ = a/h
θ = cos-1 (a/h)
θ = cos-1 (1125/1875)
θ = 53.13°
Example 5:
The hypotenuse of a right triangle is 1875 ohms(Ω). If side across from a non-right angle is 1500 ohms, how big is the angle?
Because we are given 2 sides and need to find an angle, we will have to use the inverse trig functions. Remember to label your sides first
θ = ? hypotenuse (h) = 1875 ohms opposite (a) = 1500 ohms
sinθ = o/h
θ = sin-1 (o/h)
θ = sin-1 (1500/1875)
θ = 53.13°
Example 6:
If side across from a non-right angle is 1500 ohms and the side between this angle and the right angle is 1125 ohms, how big is the angle?
Because we are given 2 sides and need to find an angle, we will have to use the inverse trig functions. Remember to label your sides first
θ = ? adjacent (a) = 1125 ohms opposite (a) = 1500 ohms
tanθ = o/a
θ = tan-1 (o/a)
θ = tan-1 (1500/1125)
θ = 53.13°
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