Introduction to Trigonometry and Its Applications
Introduction to Trigonometry and Its Applications tests ratios in right triangles, standard-angle values, identities, and heights and distances. The ratios are defined with respect to an acute angle in a right triangle.
Core rules
- sin θ = opposite/hypotenuse; cos θ = adjacent/hypotenuse; tan θ = opposite/adjacent.
- cosec θ = 1/sin θ; sec θ = 1/cos θ; cot θ = 1/tan θ.
- Identities: sin^2θ + cos^2θ = 1; 1 + tan^2θ = sec^2θ; 1 + cot^2θ = cosec^2θ.
- Standard values include sin 30°=1/2, cos 60°=1/2, tan 45°=1, sin 60°=sqrt(3)/2.
- Angle of elevation is measured upward from the horizontal; angle of depression is measured downward from the horizontal.
Worked example
Given tan θ = 3/4, find sin θ and cos θ for an acute θ. Draw a right triangle with opposite 3 and adjacent 4. The hypotenuse is sqrt(3^2+4^2)=5. Therefore sin θ = 3/5 and cos θ = 4/5. Check the identity: (3/5)^2+(4/5)^2=9/25+16/25=1.
Traps
Do not confuse opposite and adjacent sides when the triangle is rotated. The angle of depression from a tower is measured from the horizontal at the top, not from the vertical. Use reciprocal identities correctly: sec is not 1/sin. In height problems, draw a clear diagram and mark the right angle.