Finding Missing Sides and Angles

16.2: Finding Missing Sides and Angles. Original editorial SAT preparation with three worked examples, ten flashcards, and thirty practice questions. Educator-review draft.

Lesson 2 of 53 parts Practice Flashcards

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Missing leg from an exact angle

Objective: Use an exact trigonometric ratio to find a side.

Prerequisite skills: sine, cosine, tangent, and solving proportions. Choose a right-triangle ratio that relates the known side to the unknown. For standard angles, exact values such as sin30°=1/2 avoid decimal approximations. Common error: using an angle number as though it were the trig ratio. Restrictions and interpretation: Use degree mode for degree angles and radian mode for radian angles; inverse-trig angle recovery is a supporting extension.

Formulas

$sin30°=1/2; cos60°=1/2$

Worked examples

Try itA right triangle has hypotenuse 50 and an acute angle of 30°. What is the length opposite that angle?Show answer

Use sin30°=opposite/hypotenuse=1/2. Thus opposite=50/2=25. Therefore the requested answer is 25.

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Missing leg from an exact angle

Objective: Use an exact trigonometric ratio to find a side.

Formulas

$sin30°=1/2; cos60°=1/2$

Worked examples

Try itA right triangle has hypotenuse 50 and an acute angle of 30°. What is the length opposite that angle?Show answer

Use sin30°=opposite/hypotenuse=1/2. Thus opposite=50/2=25. Therefore the requested answer is 25.

Missing hypotenuse

Objective: Divide by the known side-to-hypotenuse ratio.

If a known leg is the numerator of sine or cosine, isolate the hypotenuse by division. Check the resulting hypotenuse exceeds either leg. Do not multiply by a ratio less than one to obtain the longest side. Common error: multiplying the known leg by cosine to get the hypotenuse. Restrictions and interpretation: Use degree mode for degree angles and radian mode for radian angles; inverse-trig angle recovery is a supporting extension.

Formulas

$hypotenuse=adjacent/cos θ$

Worked examples

Try itIn a right triangle, the side adjacent to acute θ is 96, and cos θ=4/5. What is the hypotenuse?Show answer

Set 96/h=4/5. Multiply by 5h: 480=4h, so h=120. This exceeds the adjacent side 96. Therefore the requested answer is 120.

Angle recovery and mode

Objective: Use an inverse function or a standard exact ratio to find an angle.

An inverse trigonometric function converts a side ratio back to an angle within a specified range. For an acute right-triangle angle, ratios such as tanθ=1 and sinθ=1/2 identify 45° and 30°. Calculator mode determines whether an inverse output is in degrees or radians. Common error: reading a radian output as a degree measure. Restrictions and interpretation: Use degree mode for degree angles and radian mode for radian angles; inverse-trig angle recovery is a supporting extension. Desmos and method choice: Use degree mode when angles are stated in degrees; confirm the mode before sin, cos, tan, or inverse calculations. Prefer exact special-angle ratios when available. A radian output such as π/4 represents 45°, not about 0.785 degrees.

Formulas

$θ=arctan(opposite/adjacent), 0°<θ<90°$

Worked examples

Try itA right triangle has equal legs of length 26. What is either acute angle in degrees?Show answer

Equal legs give tanθ=1 and make the two acute angles equal. Their sum is 90°, so each is 45°. An inverse tangent in degree mode also returns 45°. Therefore the requested answer is 45°.

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