Sin Cos Tan
Summary
TLDRThis video tutorial simplifies the understanding of the six basic trigonometric functions using the mnemonic 'sohcahtoa'. It breaks down the sine, cosine, and tangent by relating them to the sides of a right-angled triangle, labeling the hypotenuse, opposite, and adjacent sides accordingly. The script then calculates these functions for a given angle 'a', and demonstrates how to derive the remaining three functions—cosecant, secant, and cotangent—by inverting the ratios of the initial three, providing a clear and memorable method for students to grasp these mathematical concepts.
Takeaways
- 📚 The video teaches the six basic trigonometric functions using the mnemonic 'sohcahtoa'.
- 📝 'Soh' stands for sine, which is the ratio of the opposite side to the hypotenuse.
- 📝 'Cah' stands for cosine, representing the ratio of the adjacent side to the hypotenuse.
- 📝 'Toa' stands for tangent, which is the ratio of the opposite side to the adjacent side.
- 📐 The video demonstrates how to label the sides of a right triangle for trigonometric calculations.
- 📏 The hypotenuse is the longest side and is found opposite the 90-degree angle.
- 📏 The opposite side is the one directly opposite the angle in question.
- 📏 The adjacent side is the one touching the angle but not the hypotenuse.
- 🔢 The sine of angle 'a' is calculated as the opposite side (3) over the hypotenuse (5), resulting in 3/5.
- 🔢 The cosine of angle 'a' is the adjacent side (4) over the hypotenuse (5), resulting in 4/5.
- 🔢 The tangent of angle 'a' is the opposite side (3) over the adjacent side (4), resulting in 3/4.
- 🔄 Cosecant is the reciprocal of sine, so if sine is 3/5, cosecant is 5/3.
- 🔄 Secant is the reciprocal of cosine, so if cosine is 4/5, secant is 5/4.
- 🔄 Cotangent is the reciprocal of tangent, so if tangent is 3/4, cotangent is 4/3.
Q & A
What are the six basic trigonometric functions?
-The six basic trigonometric functions are sine (sin), cosine (cos), tangent (tan), cosecant (csc), secant (sec), and cotangent (cot).
What is the mnemonic 'sohcahtoa' used for?
-The mnemonic 'sohcahtoa' is used to help remember the equations for the sine, cosine, and tangent functions, where 'S' stands for sine, 'O' for opposite, 'H' for hypotenuse, 'C' for cosine, 'A' for adjacent, and 'T' for tangent.
How is the sine of an angle calculated in a right triangle?
-The sine of an angle in a right triangle is calculated as the ratio of the length of the side opposite the angle to the length of the hypotenuse.
What is the formula for cosine in the context of the provided script?
-The cosine of an angle is the ratio of the length of the adjacent side to the length of the hypotenuse.
How do you find the tangent of an angle in a right triangle?
-The tangent of an angle is found by dividing the length of the side opposite the angle by the length of the adjacent side.
What is the hypotenuse in a right triangle?
-The hypotenuse is the longest side of a right triangle, which is opposite the right angle.
How do you label the sides of a right triangle when finding trigonometric functions?
-You label the hypotenuse with 'hyp', the opposite side with 'OPP', and the adjacent side with 'adj'.
What is the cosecant of an angle and how is it related to the sine function?
-The cosecant of an angle is the reciprocal of the sine function, meaning if the sine is the ratio of the opposite side to the hypotenuse, the cosecant is the ratio of the hypotenuse to the opposite side.
How do you calculate the secant of an angle?
-The secant of an angle is the reciprocal of the cosine function, which is the ratio of the hypotenuse to the adjacent side.
What is the cotangent of an angle and how is it found?
-The cotangent of an angle is the reciprocal of the tangent function, which is the ratio of the adjacent side to the opposite side.
In the script, what is the length of the hypotenuse when finding the trigonometric functions for angle a?
-In the script, the length of the hypotenuse is given as 5 units.
What are the lengths of the sides opposite and adjacent to angle a in the script's example?
-In the script's example, the length of the side opposite angle a is 3 units, and the length of the adjacent side is 4 units.
Outlines
📚 Introduction to Basic Trigonometric Functions
This paragraph introduces the topic of the video, which is to explain the six basic trigonometric functions using the mnemonic 'sohcahtoa'. The speaker emphasizes the importance of memorizing these functions and suggests using the phrase to remember the equations for sine (opposite/hypotenuse), cosine (adjacent/hypotenuse), and tangent (opposite/adjacent). The paragraph sets the stage for a practical example that will follow.
📐 Labeling Triangle Sides for Trigonometric Functions
The speaker proceeds to demonstrate how to label the sides of a right-angled triangle in relation to a given angle 'a'. The hypotenuse is identified by its position opposite the 90-degree angle and is labeled 'hyp'. The 'opposite' side to angle 'a' is labeled 'OPP', and the 'adjacent' side is labeled 'adj'. This labeling process is crucial for calculating the trigonometric functions of angle 'a'.
🧮 Calculating Basic Trigonometric Functions
The paragraph details the calculation of the sine, cosine, and tangent of angle 'a' using the labeled sides of the triangle. The sine of 'a' is calculated as the ratio of the opposite side to the hypotenuse (3/5), the cosine as the ratio of the adjacent side to the hypotenuse (4/5), and the tangent as the ratio of the opposite side to the adjacent side (3/4). These calculations are fundamental to understanding the basic trigonometric functions.
🔄 Deriving Reciprocal Trigonometric Functions
Building upon the basic trigonometric functions, the speaker explains how to find the reciprocal functions: cosecant, secant, and cotangent. The cosecant is the reciprocal of the sine, the secant of the cosine, and the cotangent of the tangent. The paragraph provides the specific values for these functions based on the previously calculated sine, cosine, and tangent values, completing the explanation of the six basic trigonometric functions.
Mindmap
Keywords
💡Trigonometric Functions
💡SOHCAHTOA
💡Hypotenuse
💡Opposite Side
💡Adjacent Side
💡Sine
💡Cosine
💡Tangent
💡Cosecant
💡Secant
💡Cotangent
Highlights
Introduction to the six basic trigonometric functions and the mnemonic 'sohcahtoa' for memorization.
Explanation of how 'soh' in 'sohcahtoa' stands for 'sine', 'opposite', and 'hypotenuse'.
Demonstration of the 'cough' part of 'sohcahtoa' for remembering the cosine function formula.
Illustration of the 'Toa' mnemonic for the tangent function with 'opposite', 'adjacent'.
Step-by-step labeling of the sides of a right triangle for trigonometric calculations.
Identification of the hypotenuse as the longest side opposite the 90-degree angle.
Method to label the opposite side of the angle for which trigonometric functions are being found.
Description of how to label the adjacent side, touching the angle but not the hypotenuse.
Calculation of sine using the formula opposite/hypotenuse with given side lengths.
Calculation of cosine using the formula adjacent/hypotenuse with the given triangle.
Finding the tangent of angle 'a' using the formula opposite/adjacent.
Derivation of the cosecant function by inverting the sine ratio.
Explanation of finding the secant by inverting the cosine ratio.
Method to calculate the cotangent by inverting the tangent ratio.
Emphasis on the ease of finding the reciprocal trigonometric functions once the basic functions are known.
Practical application of the mnemonic 'sohcahtoa' in solving trigonometric problems.
Visual demonstration of labeling and calculating trigonometric functions for a given angle in a triangle.
Transcripts
so welcome to my video on the six basic
trig functions in this example I'm going
to find the value for the six basic trig
functions for this angle a and many
people have trouble memorizing the
equations for the basic trig functions
and one way that's easy to memorize the
equations is using the phrase so Chi Toa
I know it sounds really weird but it
helps out a lot
the phrase is sohcahtoa and if you look
at the equation for the sine you see
that the sine is equal to the opposite
over the hypotenuse and you can use the
word so to help you remember this
equation because the S stands for the
sine the O stands for the opposite and
the H stands for the hypotenuse and you
can do the same thing for the cosine in
the equation the cosine is equal to the
adjacent over the hypotenuse you can use
the word cough to help you remember that
the C stands for the cosine the a stands
for the adjacent and the H stands for
the hypotenuse and I think you can see
the pattern by now you can do the same
thing for the word Toa the T stands for
tangent the O stands for the opposite
and the a stands for the adjacent so
tangent is equal to opposite over
adjacent so let's get started right away
with this example we want to find the
values of the basic trig functions for
this angle a and the first thing that I
like to do is I like to label all the
sides of the triangle and the first side
that I always like to label is the
hypotenuse the hypotenuse is the longest
side and the easiest way to find the
hypotenuse is to go to the 90-degree
angle and if you draw an arrow to the
opposite side of the 90-degree angle
that side is always going to be your
hypotenuse so now I'm going to label the
hypotenuse with an hyp so our side with
five is the hypotenuse so now I'm going
to label the opposite side the opposite
side is always opposite of the angle
that we're trying to find so since we're
trying to find the trig functions for
angle a I'm going to go to angle a now
I'm going to draw an arrow to the
opposite side so our side with the
length of three is going to be the
opposite of angle a so now I'm going to
label side three with an OPP just to
show that it's the opposite side and
lastly we need to label the adjacent
side and the adjacent side is always
touching the angle but not the
hypotenuse so notice this side that I'm
calling over in red is touching our
angle a but it's not the hypotenuse so
that's our adjacent side the side with
the length of four is going to be our
adjacent side and I'm going to label the
side with an adj just to show that it's
the adjacent side and after we label all
the sides of the triangle it's really
easy to find the values for all the trig
functions for our angle a so that's what
I'm going to do right now first I'm
going to start off with the sine of a
and if we go back to our equation the
sine is equal to the opposite over the
hypotenuse so since our opposite is
equal to three and our hypotenuse is
equal to five the sine of a is equal to
3 over 5 so now I'll do the cosine of a
if we go back to our equation the cosine
is equal to the adjacent over the
hypotenuse since the adjacent has the
length of 4 and the hypotenuse has a
length of 5 the cosine of a is equal to
4 over 5 so now do the tangent of a we
go back to our equation the tangent is
equal to the opposite over the adjacent
since our opposite side has a length of
3 in our jacent side has a length of 4
the tangent of a is equal to 3 over 4
and once you have these
basic trig functions it's really easy to
find the last three basic trig functions
so now I'll do the cosecant of a and the
only thing you need to do to find the
cosecant of a is flip the sign of a
since the sine of a is three over five
the cosecant of a is five over three and
now I'm going to find the secant of a
and the only thing you need to do to
find a secant is flip the cosine since
the cosine of a is four over five the
secant of a is five over four and
finally we need to find the cotangent of
a in order to find the cotangent you
just need to flip the tangent so since
the tangent of a is three over four the
cotangent of a is four over three
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