Compound angles formulas

• • • • • • Compound Angle Calculator A table saw or compound miter saw can cut workpieces with two angle settings; bevel and miter. Such a saw is useful when building for example boxes with slanted sides or concrete forms for post caps. It is surprisingly complex to compute compound angle settings. On this page I have collected a few compound angle calculators that will help compute compound angles. • • • • • • • • • • Definitions: These definitions clarify the concepts discussed on this page. • A compound cut consists of two angles, the bevel angle and the miter angle. • The bevel angle (or blade tilt) is the tilt of the saw blade from vertical on the saw table. This means that a normal square cut has a bevel of 0°. Typically saws have a maximum bevel of 45°. • The miter angle (or cross-cut angle) is the horizontal angle, as seen on the saw table, from a line perpendicular to the long edge of a board. The miter angle is set on the miter gauge of the table saw. A perpendicular cut has a miter of 0°. • Some saws label the miter angle differently, with a perpendicular cut labeled 90°. This is the miter • The dihedral angle is the angle between two surface planes. Specifically, we are computing the inside dihedral angle, which is always ≤180°. A block that fits snugly between the two surfaces can be cut with the miter angle set to the dihedral angle minus 90° and with zero blade tilt. • A miter joint joins the cut ends of two boards. • A butt joint joins the cut end of one b...

Sin(a - b) Sin(a - b) is one of the important trigonometric identities used in trigonometry, also called sin(a - b) compound angle formula. Sin (a - b) identity is used in finding the value of the sine trigonometric function for the difference of given angles, say 'a' and 'b'. The expansion of sin (a - b) can be applied to represent the sine of a compound angle(in form of a difference of two angles) in terms of sine and cosine trigonometric functions. Let us understand the sin(a - b) identity and its proof in detail in the upcoming sections. 1. 2. 3. 4. 5. Proof of Sin(a - b) Formula The expansion of sin(a - b) formula can be proved geometrically. To give the stepwise derivation of the formula for the sine To prove: sin (a - b) = sin a cos b - cos a sin b Construction: Let OX be a rotating line. Rotate it about O in the anti-clockwise direction to form the rays OY and OZ such that∠XOZ = a and ∠YOZ = b. Then ∠XOY = a - b. Take a point P on the ray OY, and: • draw perpendiculars PQ and PR to OX and OZ respectively. • Again, draw Proof: We will see how we have written∠TPR = a in the above figure. • From the right triangle OPQ,∠OPQ = 180 - (90 + a - b) = 90 - a + b; • From the right triangle OPR,∠OPR = 180 - (90 + b) = 90 - b Now, from the figure,∠OPQ,∠OPR, and∠TPR are the angles at a point on a straight line and hence they add up to 180 degrees. ∠OPQ +∠OPR +∠TPR = 180 (90 - a + b) + (90 - b) +∠TPR = 180 180 - a +∠TPR = 180 ∠TPR = a Now, from the right-angled triangle PQO we g...

Sin(a + b) Sin(a + b) is one of the important trigonometric identities used in trigonometry. It is one of sum and difference formulas. It says sin (a + b) = sin a cos b + cos a sin b. We use the sin(a + b) identity to find the value of the sine trigonometric function for the sum of angles. The expansion of sin a plus b formula helps in representing the sine of a compound angle in terms of sine and cosine trigonometric functions. Let us understand the sin(a+b) identity and its proof in detail in the following sections. 1. 2. 3. 4. Proof of Sin(a + b) Formula The proof of expansion of sin(a + b) formula can be done geometrically. Let us see the stepwise derivation of the formula for the To prove: sin (a + b) = sin a cos b + cos a sin b Construction: Assume a rotating line OX and let us rotate it about O in the anti-clockwise direction. OX makes out an acute ∠XOY = a, from starting position to its initial position. Again, the rotating line rotates further in the same direction and starting from the position OY, thus making out an acute angle given as, ∠YOZ = b. ∠XOZ = a + b < 90°. On the bounding line of the compound angle (a + b) take a point P on OZ, and draw PQ and PR perpendiculars to OX and OY respectively. Again, from R draw perpendiculars RS and RT upon OX and PQ respectively. Proof: From triangle PTR we get, ∠TPR = 90° - ∠PRT = ∠TRO = alternate ∠ROX = a. Now, from the right-angled triangle PQO we get, sin (a + b) = PQ/OP = (PT + TQ)/OP = PT/OP + TQ/OP = PT/OP + RS/OP ...

Addition formulae When we add or subtract angles, the result is called a compound angle. For example, \(30^\circ + 120^\circ\) is a compound angle. Using a calculator, we find: \[\sin (30^\circ + 120^\circ ) = \sin 150^\circ = 0.5\] \[\sin 30^\circ + \sin 120^\circ = 1.366\,(to\,3\,d.p.)\] This shows that \(\sin (A + B)\) is not equal to \(\sin A + \sin B\) . Instead, we can use the following identities: \[\sin (A + B) = \sin A\cos B + \cos A\sin B\] \[\sin (A - B) = \sin A\cos B - \cos A\sin B\] \[\cos (A + B) = \cos A\cos B - \sin A\sin B\] \[\cos (A - B) = \cos A\cos B + \sin A\sin B\] • Addition formulae are given in a condensed form: • \[\sin (A \pm B) = \sin A\cos B \pm \cos A\sin B\] • \[\cos (A \pm B) = \cos A\cos B \mp \sin A\sin B\] These formulae are used to expand trigonometric functions to help us simplify or evaluate trigonometric expressions of this form. See how we approach this two-part question: Question 1. By writing \(75^\circ = 45^\circ + 30^\circ\) determine the exact value of \(\sin 75^\circ\) Reveal answer down 1. \(\sin 75^\circ = \sin (45 + 30)^\circ\) Using the formula for \(\sin (A + B)\) \[= \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ\] Using exact values that you should know: \[= \frac\) Reveal answer down 2. Since \(\frac\]

Compound Angle Calculators I recently had a woodworking project with compound angles. A quick search of the web for an online calculator quickly showed that, even though there are plenty of great miter calculators out there, I wasn't able to find to for calculate a butted joint at a compound angle. After a bit of research, I discovered the possible reason why – calculating a butted joint is much more complicated than calculating a mitered joint. Thanks to this website: Compound Miter (enter values in degrees) Slope Included Angle degrees -or- sides End Angle Bevel angle Calculate Slope Horizontal run (inches) Vertical rise (inches) Slope (degrees) Compound Angle -- Butted joint Slope A Slope B Included Angle degrees -or- sides Results: Side A end angle Side B end angle Bevel angle These two calculators were designed with a tablesaw in mind, although you could use it with any tool. Just adjust angular orientation accordingly. Notes on angles: Input: Slope angle is measured from the horizontal plane. 0° is horizontal and 90° is vertical. Included angle refers to the angle the two pieces meet in plan view. The angle is measured between the two pieces – 90° is a 4-sided box, 120° is 6-sides, etc. Results: End angles are given in relation to a square end. 0° is a square ended piece, 45° is a piece cut with a 1:1 angle. End angle refers to the angle on the end of the piece when it is laying in the horizontal plane (like on the top of a tablesaw). End angle is the angle you'd set...

Compound Angle Calculators I recently had a woodworking project with compound angles. A quick search of the web for an online calculator quickly showed that, even though there are plenty of great miter calculators out there, I wasn't able to find to for calculate a butted joint at a compound angle. After a bit of research, I discovered the possible reason why – calculating a butted joint is much more complicated than calculating a mitered joint. Thanks to this website: Compound Miter (enter values in degrees) Slope Included Angle degrees -or- sides End Angle Bevel angle Calculate Slope Horizontal run (inches) Vertical rise (inches) Slope (degrees) Compound Angle -- Butted joint Slope A Slope B Included Angle degrees -or- sides Results: Side A end angle Side B end angle Bevel angle These two calculators were designed with a tablesaw in mind, although you could use it with any tool. Just adjust angular orientation accordingly. Notes on angles: Input: Slope angle is measured from the horizontal plane. 0° is horizontal and 90° is vertical. Included angle refers to the angle the two pieces meet in plan view. The angle is measured between the two pieces – 90° is a 4-sided box, 120° is 6-sides, etc. Results: End angles are given in relation to a square end. 0° is a square ended piece, 45° is a piece cut with a 1:1 angle. End angle refers to the angle on the end of the piece when it is laying in the horizontal plane (like on the top of a tablesaw). End angle is the angle you'd set...

• • • • • • Compound Angle Calculator A table saw or compound miter saw can cut workpieces with two angle settings; bevel and miter. Such a saw is useful when building for example boxes with slanted sides or concrete forms for post caps. It is surprisingly complex to compute compound angle settings. On this page I have collected a few compound angle calculators that will help compute compound angles. • • • • • • • • • • Definitions: These definitions clarify the concepts discussed on this page. • A compound cut consists of two angles, the bevel angle and the miter angle. • The bevel angle (or blade tilt) is the tilt of the saw blade from vertical on the saw table. This means that a normal square cut has a bevel of 0°. Typically saws have a maximum bevel of 45°. • The miter angle (or cross-cut angle) is the horizontal angle, as seen on the saw table, from a line perpendicular to the long edge of a board. The miter angle is set on the miter gauge of the table saw. A perpendicular cut has a miter of 0°. • Some saws label the miter angle differently, with a perpendicular cut labeled 90°. This is the miter • The dihedral angle is the angle between two surface planes. Specifically, we are computing the inside dihedral angle, which is always ≤180°. A block that fits snugly between the two surfaces can be cut with the miter angle set to the dihedral angle minus 90° and with zero blade tilt. • A miter joint joins the cut ends of two boards. • A butt joint joins the cut end of one b...

Addition formulae When we add or subtract angles, the result is called a compound angle. For example, \(30^\circ + 120^\circ\) is a compound angle. Using a calculator, we find: \[\sin (30^\circ + 120^\circ ) = \sin 150^\circ = 0.5\] \[\sin 30^\circ + \sin 120^\circ = 1.366\,(to\,3\,d.p.)\] This shows that \(\sin (A + B)\) is not equal to \(\sin A + \sin B\) . Instead, we can use the following identities: \[\sin (A + B) = \sin A\cos B + \cos A\sin B\] \[\sin (A - B) = \sin A\cos B - \cos A\sin B\] \[\cos (A + B) = \cos A\cos B - \sin A\sin B\] \[\cos (A - B) = \cos A\cos B + \sin A\sin B\] • Addition formulae are given in a condensed form: • \[\sin (A \pm B) = \sin A\cos B \pm \cos A\sin B\] • \[\cos (A \pm B) = \cos A\cos B \mp \sin A\sin B\] These formulae are used to expand trigonometric functions to help us simplify or evaluate trigonometric expressions of this form. See how we approach this two-part question: Question 1. By writing \(75^\circ = 45^\circ + 30^\circ\) determine the exact value of \(\sin 75^\circ\) Reveal answer down 1. \(\sin 75^\circ = \sin (45 + 30)^\circ\) Using the formula for \(\sin (A + B)\) \[= \sin 45^\circ \cos 30^\circ + \cos 45^\circ \sin 30^\circ\] Using exact values that you should know: \[= \frac\) Reveal answer down 2. Since \(\frac\]

Sin(a + b) Sin(a + b) is one of the important trigonometric identities used in trigonometry. It is one of sum and difference formulas. It says sin (a + b) = sin a cos b + cos a sin b. We use the sin(a + b) identity to find the value of the sine trigonometric function for the sum of angles. The expansion of sin a plus b formula helps in representing the sine of a compound angle in terms of sine and cosine trigonometric functions. Let us understand the sin(a+b) identity and its proof in detail in the following sections. 1. 2. 3. 4. Proof of Sin(a + b) Formula The proof of expansion of sin(a + b) formula can be done geometrically. Let us see the stepwise derivation of the formula for the To prove: sin (a + b) = sin a cos b + cos a sin b Construction: Assume a rotating line OX and let us rotate it about O in the anti-clockwise direction. OX makes out an acute ∠XOY = a, from starting position to its initial position. Again, the rotating line rotates further in the same direction and starting from the position OY, thus making out an acute angle given as, ∠YOZ = b. ∠XOZ = a + b < 90°. On the bounding line of the compound angle (a + b) take a point P on OZ, and draw PQ and PR perpendiculars to OX and OY respectively. Again, from R draw perpendiculars RS and RT upon OX and PQ respectively. Proof: From triangle PTR we get, ∠TPR = 90° - ∠PRT = ∠TRO = alternate ∠ROX = a. Now, from the right-angled triangle PQO we get, sin (a + b) = PQ/OP = (PT + TQ)/OP = PT/OP + TQ/OP = PT/OP + RS/OP ...

Sin(a - b) Sin(a - b) is one of the important trigonometric identities used in trigonometry, also called sin(a - b) compound angle formula. Sin (a - b) identity is used in finding the value of the sine trigonometric function for the difference of given angles, say 'a' and 'b'. The expansion of sin (a - b) can be applied to represent the sine of a compound angle(in form of a difference of two angles) in terms of sine and cosine trigonometric functions. Let us understand the sin(a - b) identity and its proof in detail in the upcoming sections. 1. 2. 3. 4. 5. Proof of Sin(a - b) Formula The expansion of sin(a - b) formula can be proved geometrically. To give the stepwise derivation of the formula for the sine To prove: sin (a - b) = sin a cos b - cos a sin b Construction: Let OX be a rotating line. Rotate it about O in the anti-clockwise direction to form the rays OY and OZ such that∠XOZ = a and ∠YOZ = b. Then ∠XOY = a - b. Take a point P on the ray OY, and: • draw perpendiculars PQ and PR to OX and OZ respectively. • Again, draw Proof: We will see how we have written∠TPR = a in the above figure. • From the right triangle OPQ,∠OPQ = 180 - (90 + a - b) = 90 - a + b; • From the right triangle OPR,∠OPR = 180 - (90 + b) = 90 - b Now, from the figure,∠OPQ,∠OPR, and∠TPR are the angles at a point on a straight line and hence they add up to 180 degrees. ∠OPQ +∠OPR +∠TPR = 180 (90 - a + b) + (90 - b) +∠TPR = 180 180 - a +∠TPR = 180 ∠TPR = a Now, from the right-angled triangle PQO we g...