Question

Derive the formula \(\cos a=\cos b \cos c+\sin b \sin c \cos A\) for an arbitrary spherical triangle with sides \(a, b, c\) and op-posite angles \(A, B, C\) on a sphere of radius 1 by dividing the triangle into two right triangles and applying the formulas of Chapter 5 .

Short answer

Question: Derive the formula connecting the sides and angles in a spherical triangle: \(\cos a = \cos b \cos c + \sin b \sin c \cos A\). Answer: The formula can be derived by dividing the given spherical triangle into two right triangles, applying the spherical cosine formula to both right triangles, using the Law of Cosines for Spherical Triangles, applying the Pythagorean relationship in right triangles, and finally combining terms to arrive at the final formula: \(\cos a = \cos b \cos c + \sin b \sin c \cos A\).

Step-by-step solution

Divide the triangle into two right triangles.

Draw an altitude from the angle \(A\) to the opposite side \(a\). This altitude will be perpendicular to side \(a\), forming two right triangles. Let's call the point where the altitude intersects side \(a\) as point \(D\). Now we have two right triangles: triangle \(ABD\) and triangle \(ACD\).

Apply the spherical cosine formula to both right triangles.

The spherical cosine formula relates the three sides and one angle of a right spherical triangle. It states that: \(\cos(c) = \cos(a) \cos(b) + \sin(a) \sin(b) \cos(\gamma)\), where \(\gamma\) is the angle between sides \(a\) and \(b\). Let's apply this formula to the right triangles \(ABD\) and \(ACD\). For triangle \(ABD\), let \(AD\) be denoted as \(a_1\) and \(BD\) as \(a_2\). We get: \(\cos(a_1) = \cos(b) \cos(a_2) + \sin(b) \sin(a_2) \cos(90^\circ - A)\). Note that \(\cos(90^\circ - A) = \sin A\). For triangle \(ACD\), let \(CD\) be denoted as \(c_1\). We get: \(\cos(c_1) = \cos(b) \cos(c) + \sin(b) \sin(c) \cos(90^\circ - A)\).

Use the Law of Cosines for Spherical Triangles.

The Law of Cosines for Spherical Triangles states that: \(\cos(a) = \cos(a_1) \cos c + \sin(a_1) \sin(c) \cos(B)\) Substitute the expression of \(\cos(a_1)\) from Step 2: \(\cos(a) = (\cos(b) \cos(a_2) + \sin(b) \sin(a_2) \sin(A)) \cos c + \sin(b) \sin(a_2) \sin(A) \sin c\)

Apply Pythagorean relationship in right triangles and simplify.

In triangle \(ABD\), we have: \(\sin^2(a_2) = 1 - \cos^2(a_2)\) Substitute this in the expression of \(\cos(a)\): \(\cos(a) = (\cos(b) \cos(a_2) + \sin(b) \sqrt{1 - \cos^2(a_2)} \sin(A)) \cos c + \sin(b) \sqrt{1 - \cos^2(a_2)} \sin(A) \sin c\) Now, notice that \(\sqrt{1 - \cos^2(a_2)} = \sin a_2\). So, we can simplify the expression to: \(\cos(a) = (\cos(b) \cos(a_2) + \sin(b) \sin(a_2) \sin(A)) \cos c + \sin(b) \sin(a_2) \sin(A) \sin c\)

Combine terms for the final formula for \(\cos a\).

Combining the terms in the equation, we get the final formula for \(\cos a\): \(\cos a = \cos b \cos c + \sin b \sin c \cos A\) This is the desired formula connecting the sides and angles of an arbitrary spherical triangle.

Key concepts

Spherical Cosine Formula

Spherical trigonometry, a branch vital for celestial navigation and geodesy, revolves around the study of spherical triangles. A key equation in this area is the spherical cosine formula. This formula presents a relationship among the angles and sides of a spherical triangle.The elegant equation states that for a right-angled spherical triangle—with angle \( \gamma \) opposite side \( c \), the sides \( a \) and \( b \) adjacent to \( \gamma \)—the cosine of side \( c \) can be calculated using the following:\[ \cos(c) = \cos(a) \cos(b) + \sin(a) \sin(b) \cos(\gamma) \]
This formula is vital in deriving relationships between any given sides and angles in a spherical triangle, not only the right-angled ones. It enables us to explore and solve complex problems by breaking them down into solvable units.

Law of Cosines for Spherical Triangles

Expanding the Spherical Cosine Formula

The Law of Cosines for Spherical Triangles expands upon the spherical cosine formula. It articulates the relationship between the lengths of sides of a spherical triangle and the cosine of one of its angles. More specifically, this law is given for an arbitrary spherical triangle by the formula:\[ \cos(a) = \cos b \cos c + \sin b \sin c \cos A \]
Here, the sides \( a \) , \( b \) , and \( c \) form a spherical triangle with their respective opposite angles \( A \) , \( B \) , and \( C \) . This law makes it feasible to compute the length of any side of a spherical triangle when the other two sides and the included angle are known. Its crucial role in solving non-right spherical triangles echoes in various fields such as astronomy and geodetic science.

Spherical Triangles

Imagine a triangle drawn not on a flat plane, but curved over the surface of a sphere: this is a spherical triangle. Each vertex of a spherical triangle lies on the surface of the sphere, and the sides are arcs of great circles—the largest possible circles that can be drawn on a sphere.

Characteristics Unique to Spherical Triangles

Unlike their flat counterparts, the sum of the angles in a spherical triangle exceeds 180 degrees, and the sides are not straight lines but curve with the sphere's surface. The interesting property of spherical triangles is that they are defined by their angles, which, in turn, measure the portion of the spherical surface they encompass.

• The greater the sum of the triangle's angles, the larger the area it covers on the sphere.
• Each angle can be greater than 90 degrees, unlike in plane triangles.
• Understanding the spherical triangle properties is essential for applying the spherical cosine formula and the Law of Cosines.

Mathematical Proof

Proving Relationships in Spherical Trigonometry

In spherical trigonometry, a mathematical proof serves to validate the relationships between the angles and sides of spherical triangles. The proof for the spherical cosine formula, for instance, hinges on the methodical application of right-angled triangle principles and the Law of Cosines for Spherical Triangles.Proof typically begins by dividing the spherical triangle into two right-angled triangles. This step simplifies the problem and employs the spherical cosine formula for right-angled triangles. Progressively, through algebraic manipulation and application of trigonometric identities, the general formula is proven.
The proof elucidates the correctness of these relationships, assuring that the formulas used are founded on solid mathematical principles. It's an indispensable technique in mathematics to not only establish the veracity of formulas but to also deepen understanding and ensure accurate application in various scenarios such as navigation and astronomy.