This section requires a unit circle and table. Use the one from last section or print one below! Show
In this worksheet, we will practice finding the trigonometric function values for 30-, 45-, and 60-degree angles. Q1: Find the exact value of sin 30∘.
Q2: Find tan 30∘.
Q3: Find the value of 1+603030tantancos∘∘∘.
Q4: Find the value of 24530 cossin∘∘.
Q5: Find the value of cossinsintantan6030 −6060+30∘∘∘∘∘ without using a calculator.
Q6: Find the value of tantansincos601−453060∘∘∘∘ without using a calculator.
Q7: If cos𝑥=12, find the value of 𝑥, where 0<𝑥 <90∘. Q8: If 2𝑥=1sin, find the value of 𝑥, where 0<𝑥<90∘. Q9: If √3𝑥+2=3tan, find the value of 𝑥, where 0<𝑥<90∘. Q10: Given that sincos𝑥60=14 , find the value of 𝑥, where 0< 𝑥<90∘. This lesson includes 47 additional questions and 266 additional question variations for subscribers. Trigonometry is known as the branch of mathematics that deals with certain measurements of triangular regions. A common application of trigonometry is the measurement of the sides and angles of a triangle. For this, we use some trigonometric functions of acute angles. These functions are defined as certain ratios of a right-angled triangle containing the angle. The values of trigonometric ratios of certain angles, called standard angles,can be obtained geometrically. These angles are 0°, 30° or π/6, 45° or π/4, 60° or π/3 and 90° or π/2. Trigonometric IdentitiesIn mathematics, trigonometric identities are equalities that involve trigonometric functions and are true for each value of the occurring variables that each side of the equality is defined. Co-function Identities
Supplement Angle Identities
Values of Trigonometric Ratios of 45°. Let ABC be a right-angled isosceles triangle in which ∠ABC = 90° and AB = BC. From geometry, ∠ABC = ∠BAC = 45°. If BC = k then AB = k. ∴ AB2 + BC2 = AC2 (By Pythagoras Theorem) ∴ k2 + k2 = AC2, or AC2 =2k2. ∴ AC = √2 k. Now, sin 45° = sin C= p/h = AB/AC = k/√2 k = 1/√2 cos 45° = cos C = b/h = BC/AC = k/√2 k = 1/√2 tan 45°= tan C= p/b = AB/BC = k/ k = 1 cosec, sec, cot being the reciprocal of sin, cos, tan respectively will have just the reciprocal values as follows cosec 45°= √2 , sec 45° = √2 and cot 45° = 1. Values of Trigonometric Ratios of 30° and 60° Let ABC be an equilateral triangle whose each side is k. By geometry, each angle of the triangle = 60°. Let AD⊥BC. From geometry, AD bisects ∠BAC and it also bisects the side BC. ∴ ∠CAD = ∠BAD = 30° and CD = BD = k/2. In the right-angled △ADC, AD2 + DC2 = AC2 AD + k2/2 = k2 OR AD = k2 – k2/4 = 3k2/4. AD = √3k/2. sin 30° = sin∠CAD = p/h = CD/AC = (k/2)/k = 1/2 cos 30° = cos∠CAD = b/h = AD/AC = (√3k/2)/k = √3/2 tan 30° = tan∠CAD = p/b = CD/AD = (k/2)/(√3k/2) = 1/√3 As shown earlier, cosec 30°, sec 30°, cot 30° being the reciprocal of sin, cos, tan respectively will have just the reciprocal values. sin 60° = sin∠CAD = p/h = AD/AC = (√3k/2)/k = √3/2 cos 60° = cos∠CAD = b/h = CD/AC = (k/2)/k =1/2 tan 60° = tan∠CAD = p/b = AD/CD = (√3k/2)/(k/2) = √3 Also, cosec 60°, sec 60°, cot 60° being the reciprocal of sin, cos, tan respectively will have just the reciprocal values. Values of Trigonometric Ratios of 0° and 90° In a right-angled triangle the measure of no angle can be 0°, and neither can there be another angle of 90°. As we have seen, the triangle ratios of θ (when 0° < θ < 90°) can be obtained from their definitions. The values of trigonometric ratios turn out as follows. sin 90°= 0, cos 0° = 1, tan 0° = 0, sec 0° = 1, sin 90° = 1, cos 90° = 0, cot 90° = 0, sec 90° = 1. Other trigonometric ratios of 0° and 90° are not defined. Tabulated values
How to evaluate trigonometry functions without a calculator?As known, there are four quadrants in trigonometry, the first quadrant being all positive trigonometric values, the second quadrant is where only sine and cosec are positive, in the third quadrant only tan and cot are positive, and in the fourth one cosine and sec are positive. (Go Anti-clockwise from Right-hand Top). The above given trigonometric ratios of standard values, as well as the trigonometric identities, will help us to find an angle in trigonometry without a calculator. If sin 150° is given to find out, we can write or elaborate this as, Steps
Note The standard values should be memorized. Sample ProblemsQuestion 1: Find tan 135° without using a calculator. Solution:
Question 2: Find cos 330°. Solution:
Question 3: Find sec 120°. Solution:
Question 4: Find sin 390°. Solution:
Question 5: Find cot 150°. Solution:
How do you evaluate trig functions without a calculator?For example, if you are given a triangle whose angles are 90 degrees, 12 degrees and 78 degrees, the hypotenuse (the side opposite the 90-degree angle) is 24, and the side opposite the 12-degree angle is 5. You would therefore divide the opposite side by the hypotenuse, 5/24, to get 0.21 as the sine of 12 degrees.
|