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Trigonometry in relation to angles and triangles

5 min read

From: 3blue1brown

Trigonometry is fundamentally about understanding the relationship between angles and the sides of triangles, particularly right triangles [00:20:32]. While often introduced through triangles, a deeper understanding reveals its core connection to circles [00:13:18].

Introduction to Trigonometric Functions

Trigonometric functions, such as cosine and sine, describe cycling phenomena and are relevant in fields like physics for studying waves [00:00:57]. By manipulating their inputs and outputs on a graphing calculator, one can observe how these functions behave [00:01:07]. For instance, putting a larger constant in front of x in cosine(x) squishes the graph, increasing its frequency [00:01:10].

A surprising discovery from graphing cosine(x) is that squaring cosine(x) (i.e., cos²(x)) results in a graph that looks like a scaled-down, faster-oscillating cosine wave [00:04:47]. This non-obvious behavior suggests a deeper mathematical relationship, particularly a connection to exponential functions [00:12:31]. For example, (2^x)² is equivalent to 2^(2x), showing a similar property where squaring a function is like scaling its input [00:12:17].

Understanding Sine and Cosine Using Unit Circles

A powerful way to understand sine and cosine is through the unit circle [00:13:51]. Imagine starting at the rightmost side of a circle with a radius of one [00:14:07]. As you walk around the circle at a constant rate:

  • Sine (sin θ): The sine of an angle (or distance walked) represents your height, or y-coordinate, on the unit circle [00:14:15].
  • Cosine (cos θ): The cosine of an angle represents your x-coordinate, or distance from the vertical line [00:15:12].

Since a full walk around the unit circle is 2π units (where π units gets you halfway around), trigonometric functions naturally cycle [00:14:28]. For instance, cosine(3) is approximately -0.99 and sine(3) is approximately 0.14, because walking 3 units around the circle places you just shy of halfway around (π ≈ 3.14) [00:18:33].

Radians vs. Degrees

In mathematics, angles are often expressed in radians, which represent the literal distance walked along the circumference of a unit circle [00:15:50]. This is considered a more “natural unit” than degrees, as it directly relates to arc length [00:27:17]. For example, 180 degrees is equivalent to π radians [00:29:38]. When dealing with calculus involving trigonometric functions, radians are almost always preferred [00:30:30].

Trigonometric Functions in Right Triangles (SOH CAH TOA)

The classic way to remember trigonometric definitions for a right triangle is “SOH CAH TOA” [00:20:42]:

For example, if a 100-meter leaning tower makes an 80-degree angle with the ground, the length of its shadow when the sun is directly overhead (forming a right triangle) would be 100 * cosine(80 degrees) [00:25:49].

This triangle-based definition connects to the unit circle by scaling any right triangle so its hypotenuse is 1 [00:27:44]. In such a triangle, the opposite side becomes sin θ and the adjacent side becomes cos θ, representing the y and x coordinates on a unit circle, respectively [00:28:08].

Computing Specific Values and Identities

Computing exact values for sine and cosine can be challenging [00:30:52]. However, some values can be derived using geometric symmetries. For example, by bisecting an equilateral triangle (with side lengths of 1), a 30-60-90 right triangle is formed [00:31:29].

The Pythagorean Identity

A fundamental trigonometric identity, directly derived from the Pythagorean theorem on a unit circle, is cos²(x) + sin²(x) = 1 [00:39:31]. This means if you know one function, you can find the other [00:39:40].

  • Note on Notation: In trigonometry, cos²(x) is a shorthand for (cos(x))², which differs from how squaring a function f(x) is usually written in other areas of mathematics (f²(x) often means f(f(x))) [00:40:04].

Half-Angle Identity

The relationship discovered by graphing cosine(x) squared, cos²(x) = (1 + cos(2x)) / 2, is a version of the half-angle identity [00:48:24]. This identity allows for the computation of values that are otherwise difficult to determine, such as cosine(π/12) (15 degrees) [00:48:43]. By plugging x = π/12 into the identity: cos²(π/12) = (1 + cos(2 * π/12)) / 2 cos²(π/12) = (1 + cos(π/6)) / 2 Since cos(π/6) = √3 / 2: cos²(π/12) = (1 + √3 / 2) / 2 cos(π/12) = √((1 + √3 / 2) / 2) [00:54:19]

This identity highlights the connection between trigonometry and complex numbers, where doubling the angle corresponds to a squaring operation [00:56:15].

Geometric Interpretation of Tangent

The tangent function (tan θ) can also be visualized on the unit circle [00:56:44]. If a line tangent to the unit circle is drawn at the point (1,0) and extended upwards or downwards to intersect the terminal side of the angle θ (extended if necessary), the length of this tangent segment is tan θ [00:58:11]. This is because a right triangle can be formed where the adjacent side is 1 (the radius) and the opposite side is tan θ [00:58:15].

  • As θ approaches π/2 (90 degrees), tan θ approaches infinity, as the vertical line from the origin becomes parallel to the tangent line at (1,0) [01:03:08].

Geometric Interpretation of cos²(θ) and sin²(θ)

Within the unit circle triangle (where the hypotenuse is 1, and legs are cos θ and sin θ), a powerful visual proof of the Pythagorean identity exists [01:05:24]. By drawing a perpendicular line from the vertex of the right angle to the hypotenuse, two smaller right triangles are formed that are similar to the original [01:09:50].

  • The length of the segment on the hypotenuse adjacent to cos θ is cos²(θ) [01:07:12].
  • The length of the segment on the hypotenuse adjacent to sin θ is sin²(θ) [01:09:32]. Since these two segments make up the entire hypotenuse, and the hypotenuse’s length is 1, it visually demonstrates that cos²(θ) + sin²(θ) = 1 [01:09:44]. This geometric interpretation provides a deep insight into the Pythagorean theorem itself [01:09:50].

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