Lesson: Trigonometric Functions
Hello everyone! Welcome to the world of Trigonometric Functions. This chapter is one of the cornerstones of the A-Level Applied Mathematics 1 exam. Many students might feel intimidated because there are so many formulas, but if you understand the "origin" and the "relationships" behind them, I promise this will become one of the most enjoyable chapters to score points in!
In this chapter, we will shift our perspective from simple "right-angled triangles" to the concepts of circles and periodic motion, which are used in everything from explaining sound waves to the orbits of stars!
1. Fundamental Basis: The Unit Circle
If it feels difficult at first, don't worry! Just imagine a circle with a radius of 1 unit placed on an \(X\) and \(Y\) coordinate plane, with its center at \((0,0)\).
Key Point: Every point on the circumference is \((x, y)\), which we redefine as:
- The value of \(\cos \theta\) is the coordinate on the \(X\)-axis.
- The value of \(\sin \theta\) is the coordinate on the \(Y\)-axis.
- Therefore, any point on the unit circle is \((\cos \theta, \sin \theta)\).
Did you know? Since it is a circle with a radius of 1, according to the Pythagorean theorem, we get the eternal formula:
\( \sin^2 \theta + \cos^2 \theta = 1 \)
Common Pitfall: Students often get confused with the signs (+/-) in each quadrant. Just remember the rule: "All-Sin-Tan-Cos":
- Q1 (Top Right): All (Everything is positive)
- Q2 (Top Left): Sin is positive (others are negative)
- Q3 (Bottom Left): Tan is positive (others are negative)
- Q4 (Bottom Right): Cos is positive (others are negative)
Summary of this section: The unit circle is the origin of every trigonometric formula. Master the coordinates \((x, y) = (\cos, \sin)\) thoroughly!
2. Other Trigonometric Functions and Relationships
Beyond \(\sin\) and \(\cos\), there are four other "friends" created by dividing the first two or taking their reciprocals:
1. \(\tan \theta = \frac{\sin \theta}{\cos \theta}\)
2. \(\csc \theta = \frac{1}{\sin \theta}\) (Reciprocal of sin)
3. \(\sec \theta = \frac{1}{\cos \theta}\) (Reciprocal of cos)
4. \(\cot \theta = \frac{1}{\tan \theta} = \frac{\cos \theta}{\sin \theta}\)
Memory Tip: \(\csc\) starts with c and is the reciprocal of the function starting with s (\(\sin\)), while \(\sec\) starts with s and is the reciprocal of the function starting with c (\(\cos\)). They always swap!
3. Graphs of Trigonometric Functions
Trigonometric functions have a unique characteristic called being a "Periodic Function," meaning they repeat themselves in cycles, much like the swinging of a pendulum.
Graphs of \(y = \sin x\) and \(y = \cos x\)
- Amplitude: The height of the wave measured from the midline (normally 1).
- Period: The length of one full cycle (normally \(2\pi\) or \(360^\circ\)).
Important for the Exam:
If you encounter the form \(y = A \sin(Bx)\):
- The value \(|A|\) is the amplitude.
- The period is calculated as \(\frac{2\pi}{|B|}\).
Example: \(y = 3 \sin(2x)\) will have a height of 3 units and complete one full cycle at a distance of \(\pi\) (because \(\frac{2\pi}{2} = \pi\)).
4. Must-Know Trigonometric Formulas (Selected for high exam frequency!)
Don't try to memorize them all in one day. Focus on solving problems and referencing the formulas; you will naturally remember them with practice!
Group 1: Sum and Difference of Angles
\( \sin(A \pm B) = \sin A \cos B \pm \cos A \sin B \) (Mnemonic: Sin Cos Cos Sin, same sign)
\( \cos(A \pm B) = \cos A \cos B \mp \sin A \sin B \) (Mnemonic: Cos Cos Sin Sin, opposite sign)
Group 2: Double Angle Identities (Appear very frequently!)
\( \sin 2A = 2 \sin A \cos A \)
\( \cos 2A = \cos^2 A - \sin^2 A \)
(Or transform it into \(2\cos^2 A - 1\) or \(1 - 2\sin^2 A\))
Summary of this section: These formulas are your tools to "transform" difficult problems into simpler, more manageable forms.
5. Inverse Trigonometric Functions
Referred to briefly as Arc functions, such as \(\arcsin, \arccos, \arctan\), they are essentially asking the question in reverse.
Example: If \(\sin 30^\circ = 0.5\), then what is \(\arcsin(0.5)\)? The answer is \(30^\circ\).
Caution: Arc functions have restricted ranges to ensure they remain true functions:
- \(\arcsin\) and \(\arctan\) return values in the interval \([-\frac{\pi}{2}, \frac{\pi}{2}]\) (right side of the circle).
- \(\arccos\) returns values in the interval \([0, \pi]\) (top half of the circle).
6. Law of Sines and Cosines (Applications to Triangles)
Used for any triangle, not necessarily right-angled ones.
1. Law of Cosines: (Use when you know 2 sides and the included angle)
\( a^2 = b^2 + c^2 - 2bc \cos A \)
2. Law of Sines: (Use when you know side-angle pairs)
\( \frac{a}{\sin A} = \frac{b}{\sin B} = \frac{c}{\sin C} \)
Analogy: The Law of Cosines is like an "upgraded version of Pythagoras" that works even for irregular, "crooked" triangles!
Final Tips Before the Exam
1. Always draw a diagram: No matter the problem, drawing a unit circle or a triangle helps you visualize the scenario and avoid mistakes with signs.
2. Check the units: Look closely at whether the problem uses Degrees or Radians. Don't forget that \(\pi = 180^\circ\).
3. Practice basic angle values: Values for \(0^\circ, 30^\circ, 45^\circ, 60^\circ, 90^\circ\) should be second nature to you.
Key Takeaway: Trigonometry is not about memorizing groundless formulas; it’s about understanding the relationships of coordinates on a circle. If you understand the unit circle, you’ve already won half the battle! Keep at it, I'm cheering for you!