Find the exact value of the variables in each figure.

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Identify the given information in the figure, such as known side lengths and angle measures, and label the variables you need to find.

Determine which trigonometric ratios (sine, cosine, tangent) or the Pythagorean theorem are applicable based on the right triangles or angles present in the figure.

Write down the relevant trigonometric equations using the definitions: for example, \(\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}\), \(\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}\), and \(\tan \theta = \frac{\text{opposite}}{\text{adjacent}}\).

Set up equations for each variable using the known values and solve these equations step-by-step to express the variables in terms of known quantities.

Check your solutions by verifying that the values satisfy the original triangle properties, such as the sum of angles being \(180^\circ\) and the Pythagorean theorem for right triangles.

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Trigonometric Ratios

Trigonometric ratios such as sine, cosine, and tangent relate the angles of a right triangle to the ratios of its sides. Understanding these ratios allows you to find unknown side lengths or angles when given partial information about a triangle.

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Introduction to Trigonometric Functions

Solving for Unknowns in Triangles

Finding exact values of variables in geometric figures often involves setting up equations based on known angle measures and side lengths. Using trigonometric identities and ratios, you can solve these equations to determine unknown sides or angles.

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Solving Right Triangles with the Pythagorean Theorem

Exact Values of Special Angles

Certain angles like 30°, 45°, and 60° have well-known exact trigonometric values expressed in simplified radical form. Recognizing these special angles helps in finding precise values without relying on decimal approximations.

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