Trigonometric Ratios
Aligned to G-SRT.C.6 — Common Core State Standards for Mathematics.
What this lesson teaches
In a right triangle, the ratio of two side lengths depends only on the acute angle, not the triangle's size, because all right triangles with that angle are similar. This gives the definitions: sine = opposite/hypotenuse, cosine = adjacent/hypotenuse, tangent = opposite/adjacent.
Worked example
A right triangle has an acute angle t with the opposite leg 3, the adjacent leg 4, and hypotenuse 5. Compute the three ratios: sin t = opposite/hypotenuse = 3/5 = 0.6; cos t = adjacent/hypotenuse = 4/5 = 0.8; tan t = opposite/adjacent = 3/4 = 0.75. Any larger 3-4-5 triangle gives the same ratios for angle t.
Practice questions
- A right triangle has opposite 5, adjacent 12, hypotenuse 13 for angle t. Find sin t, cos t, tan t.
- Explain why two different-sized right triangles with the same acute angle give the same sine value.
- For angle t, tan t = 8/15. Draw the legs and find the hypotenuse, then give sin t.
Watch the lesson
Every lesson comes with a video taught in English and Spanish — the same video the QR code in the printed workbook opens.
▶ Watch this lessonEn español
Razones trigonométricas
En un triángulo rectángulo, la razón de dos longitudes depende solo del ángulo agudo, no del tamaño del triángulo, porque todos los triángulos rectángulos con ese ángulo son semejantes. Esto da las definiciones: seno = opuesto/hipotenusa, coseno = adyacente/hipotenusa, tangente = opuesto/adyacente.
Ejemplo: Un triángulo rectángulo tiene un ángulo agudo t con el cateto opuesto 3, el adyacente 4 e hipotenusa 5. Calcula las tres razones: sen t = opuesto/hipotenusa = 3/5 = 0.6; cos t = adyacente/hipotenusa = 4/5 = 0.8; tan t = opuesto/adyacente = 3/4 = 0.75. Cualquier triángulo 3-4-5 más grande da las mismas razones para el ángulo t.
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