Understanding the fundamental trigonometric ratios—sine, cosine, and tangent—is essential in the study of geometry and trigonometry. These ratios, defined using right-angled triangles, form the backbone of various mathematical applications in the IB Maths: AI SL curriculum. Mastery of these concepts not only aids in solving geometric problems but also in comprehending more advanced topics in mathematics and related fields.

The sine of an angle in a right-angled triangle is defined as the ratio of the length of the side opposite the angle to the length of the hypotenuse. Mathematically, for a given angle $\theta$, the sine function is expressed as: $$ \sin(\theta) = \frac{\text{Opposite Side}}{\text{Hypotenuse}} $$ **Example:** Consider a right-angled triangle where angle $\theta = 30^\circ$, the opposite side measures 5 units, and the hypotenuse measures 10 units. The sine of $30^\circ$ is: $$ \sin(30^\circ) = \frac{5}{10} = 0.5 $$ The sine function is crucial in determining unknown side lengths and angles within right-angled triangles, making it a vital tool in both academic and real-world applications.

The cosine of an angle in a right-angled triangle is the ratio of the length of the adjacent side to the length of the hypotenuse. For angle $\theta$, it is defined as: $$ \cos(\theta) = \frac{\text{Adjacent Side}}{\text{Hypotenuse}} $$ **Example:** In a right-angled triangle where angle $\theta = 60^\circ$, the adjacent side is 8 units long, and the hypotenuse is 16 units long. The cosine of $60^\circ$ is: $$ \cos(60^\circ) = \frac{8}{16} = 0.5 $$ Cosine functions are instrumental in calculating distances, heights, and in various engineering applications where angle measurements are involved.

The tangent of an angle in a right-angled triangle is the ratio of the length of the side opposite the angle to the length of the adjacent side. It is defined as: $$ \tan(\theta) = \frac{\text{Opposite Side}}{\text{Adjacent Side}} $$ **Example:** In a right-angled triangle with angle $\theta = 45^\circ$, if the opposite side measures 7 units and the adjacent side measures 7 units, the tangent of $45^\circ$ is: $$ \tan(45^\circ) = \frac{7}{7} = 1 $$ Tangent functions are particularly useful in scenarios involving slopes and inclines, such as calculating the grade of a hill or the angle of elevation.

Trigonometric ratios can be derived using the properties of right-angled triangles and the Pythagorean theorem. For a right-angled triangle with sides of lengths $a$, $b$, and hypotenuse $c$, the Pythagorean theorem states: $$ a^2 + b^2 = c^2 $$ Using this relationship, the trigonometric ratios can be expressed in terms of the sides of the triangle:

• Sine: $\sin(\theta) = \frac{a}{c}$
• Cosine: $\cos(\theta) = \frac{b}{c}$
• Tangent: $\tan(\theta) = \frac{a}{b}$

These foundational definitions allow for the exploration of more complex trigonometric identities and equations.

Trigonometric ratios have a wide range of applications in various fields:

• Engineering: Calculating forces, designing structures, and analyzing mechanical systems.
• Physics: Understanding wave functions, oscillations, and projectile motions.
• Navigation: Determining distances and plotting courses using angles.
• Architecture: Designing buildings and ensuring structural integrity through precise measurements.

In the IB Maths: AI SL curriculum, these applications are explored to provide students with practical insights into the utility of trigonometric concepts.

Several fundamental identities involve sine, cosine, and tangent functions. These identities simplify complex trigonometric expressions and are essential in solving trigonometric equations.

• Pythagorean Identity: $$ \sin^2(\theta) + \cos^2(\theta) = 1 $$
• Tangent Identity: $$ \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} $$
• Reciprocal Identities:
  • $\csc(\theta) = \frac{1}{\sin(\theta)}$
  • $\sec(\theta) = \frac{1}{\cos(\theta)}$
  • $\cot(\theta) = \frac{1}{\tan(\theta)}$

These identities are foundational in higher-level mathematics, including calculus and advanced geometry.

To solve a right-angled triangle means to find the lengths of all its sides and the measures of all its angles. Trigonometric ratios are instrumental in this process:

• Finding Unknown Sides: Given one angle and one side, use sine, cosine, or tangent to find the other sides.
  • Example: In a triangle with $\theta = 30^\circ$ and hypotenuse $c = 10$ units, find the opposite side $a$. $$ \sin(30^\circ) = \frac{a}{10} \implies a = 10 \times \sin(30^\circ) = 10 \times 0.5 = 5 \text{ units} $$
• Determining Unknown Angles: Given two sides, use inverse trigonometric functions to find the unknown angles.
  • Example: In a triangle with opposite side $a = 7$ units and adjacent side $b = 24$ units, find angle $\theta$. $$ \tan(\theta) = \frac{7}{24} \implies \theta = \tan^{-1}\left(\frac{7}{24}\right) \approx 16.26^\circ $$

Mastering these techniques is crucial for solving real-world problems involving right-angled triangles.

Trigonometric ratios are applied in various real-life scenarios:

• Surveying: Determining land elevations and distances using theodolites and trigonometric calculations.
• Architecture: Designing roofs and structures to ensure stability and aesthetic appeal.
• Computer Graphics: Creating realistic animations and simulations by calculating angles and distances.
• Medicine: Understanding body mechanics and designing prosthetics through precise measurements.

These applications demonstrate the practical importance of trigonometric ratios beyond theoretical mathematics.

• Sine, cosine, and tangent are fundamental trigonometric ratios defined using right-angled triangles.
• These ratios allow for the calculation of unknown sides and angles in geometric problems.
• Trigonometric identities and applications extend their utility to various real-world fields.
• Mastery of these concepts is essential for success in IB Maths: AI SL and beyond.